No, on the occasion in question Putin did not say that traitors would be killed. The quote in your question comes from a March 5, 2018 broadcast of BBC Newsnight. It is a concatenation of three soundbites from a three-minute statement in which Putin says that Russia no longer kills traitors. The soundbites come from the last paragraph of his statement in which Putin paints a melodramatic picture of traitors as broken men living out their remaining days in abject misery leading to an early death.
The particular traitor to whom Putin alludes is not Sergei Skripal as some have claimed. He is correctly understood to allude to Colonel Aleksander Poteyev convicted in absentia of betraying deep-cover spies in the United States.
The actual fate of enemies of the Russian state is beyond the scope of the question and this answer. Instead we will discuss the beliefs on this subject which Putin was attempting to instill. He was inviting the Russian people to picture traitors dying friendless and alone Men Without a Country, presumably in apartments strewn with empty vodka bottles.
The translation in the BBC broadcast alters the tone of Putin's statement and broadens the set of possible meanings of his words to the point that, when they are read in the style used by Western comedians portraying Putin, they seem to convey veiled threats of violence.
The next day (March 6th) the composite soundbite from the BBC broadcast appeared at the head of article on the website of The Sun, now shorn of the context provided in the BBC broadcast. The article describes the statement as a "threat to 'choke traitors'", thereby changing the BBC's poor translation into an unambiguously false one (as discussed below).
The day after that (March 7th) the Independent put up an article with its own version of the BBC video. The article incorrectly identifies it as a video which "re-emerged online" and describes Putin's words as "apparent death threats".
Also on March 7th the Telegraph described Putin's words unambiguously as a "death threat". This despite the fact that in 2010 they had reported on the very same statement and found exactly the opposite meaning in it. What is more, the Telegraph goes beyond saying that Putin's words were spoken close to the time of Sergei Skripal's release to assert that Putin was speaking of him explicitly.
On March 7th on Good Morning Britain Piers Morgan asked Alexander Nekrassov (former Kremlin adviser) what Putin meant by "kick the bucket". Mr. Nekrassov's answers are apparently interpretations of "they will croak all by themselves" which Putin can be heard saying in Russian in the BBC audio. Mr. Morgan sees this as an evasion. Mr. Nekrassov in turn sees Mr. Morgan's interpretation that Putin meant "they will be killed" as perverse and becomes annoyed. Neither one of them seemed to know the context of the quote.
On March 12th the video was mentioned in an editorial in the New York Times. The editorial links to the March 7th article in the Independent and quotes the translation from the video. The editors seem to have obtained some information about the TV show during which the statement was made, but this is a bit garbled too. In particular their description of the question is incorrect and they make no mention of the overall import of Putin's answer.
Also on March 12th Newsweek made explicit the hitherto unspoken assumption that Putin had spoken in the style of "a Mafia boss in the Godfather series" while somewhat incongruously expressing doubt that Putin had been involved in the attempted assasination.
On March 14th an emergency meeting of the U.N. Security Council was held. Vassily Nebenzia (Russian ambassador to the United Nations) compared high British officials who accused Russia of the poisoning of Sergei Skripal with Inspector Lestrade who regularly got stuck on superficially plausible theory of the crime and when it turned out to be a dead-end was unable to extricate himself without the aid of Sherlock Homes. To this Jonathan Allen, deputy ambassador to the United Nations rejoined:
An finally, my colleague quotes fiction. Let me quote the Russian president when we think about who benefits. In 2010 he said, quote "Traitors will kick the bucket, believe me. Those other folks betrayed their friends, their brothers in arms. Whatever they got in exchange for it, those thirty pieces of silver they were given, they will choke on them."
Mr. Allen can perhaps be excused for thinking the quote authentic. It had been bandied about in the press for more than a week almost unchallenged. Putin had spoken on national TV and so nothing stopped the press from going back to look at the original video.
Earlier Western Press Coverage of Putin's Statement
A preliminary verification would not even have required a knowledge of the Russian language. Putin's statement had attracted some press coverage in the West at the time he made it. A number of these reports are available online with fuller, more illuminating versions of the quote in superior translations:
RT's Coverage of Putins Statement
On December 16, 2010 the English-language Russian TV network ran a story on its website about the call-in show. They reported on Putin's "traitors" statement under the subheading "No more 'squads against traitors'".
On December 19, 2010 RT did a 60-second TV spot on Putin's question-and-answer session. It begins with two glamor photos of Anna Chapman (one of the spies deported in the Illegals affair as described below) and follows with Putin's statement about what would happen to traitors. The segment can be viewed on Archive.org.
The Illegals Affair
In the part of the Putin's statement from which the BBC took its soundbites he refers to the exposure and arrest of agents of the Illegals Program on June 27, 2010. The Illegals Program (which is the name given it by the US Department of Justice) planted Russian sleeper agents in the United States under deap cover as private citizens. They posed as ordinary people, generally married couples, living ordinary lives in suburbia. The incident was the inspiration for the TV program The Americans.
In July of 2010 the ten spies arrested in the United States were exchanged for three Russian nationals who had been convicted of high treason for espionage. One them was Sergei Skripal, included in the exchange at the request of Great Britain. (This appears to be his only connection to the affair.) In August the deported Russian spies were “warmly greeted” by Vladimir Putin who "led them in singing patriotic songs". Later in June 2016 Putin claimed a personal connection to sleeper spy cells (details below).
After the program was exposed, the Russian Foreign Intelligence Service began an investigation in an attempt to determine whether the agents had been betrayed. Suspicion fell on one Colonel Aleksander Poteyev who was in charge of undercover spying in the US. He is thought to have fled Russia a few days before the arrest of the undercover agents. Where he is now is unclear, but the opinion in the Russian press is that he went to the US, that his children are also in the US, and that he may or may not have died in 2016.
Putin's comment on the Illegals affair came at the end of a longer statement in answer to a question on another subject. He is speaking during the program Direct Line: A Conversation with Vladimir Putin in December 2010. Direct Line is a marathon ask-me-anything-style show which he does once a year. The broadcast is on Youtube (the question is asked at about 3:12:15 and Putin concludes his answer at about 3:15:15) and there is a written transcript. Here is an English translation. The parts included in some form in the quote from your question are in bold:
M. Sittel: Vladimir Vladimirovich, I’m taking a question from the website, this time it is one of those personal ones. It is clearly written by someone who loves memoirs. “When you spoke of the recent spy scandal, you noted: traitors do not live long. The leaders of many countries, as know from recollections, have signed orders for the the liquidation of the enemies of the homeland oversees. The French have done so, the Israelis. Have you, as the head of state, had occasion to make such a decision in the past?”
V. Putin: I do not think that leaders of state signed such orders personally even in the past. That is the work of the special services. And during Soviet times, in Stalin’s time, it is no secret, there were special subunits which carried out, including (these were military subunits, it was not all they did), which when necessary carried out such assignments: the liquidation of traitors. Such subunits were themselves liquidated long ago.
It is known that actually many, say the Israeli special services have (pause) used such methods. Yes, all things considered, as for today, far from all have given this up even now. The Russian special services do not use such means.
With regard to traitors, they will curl up on their own, I assure you. That’s because...because... Well picture for yourself, take this latest instance of betrayal in which they exposed a group of our illegals. And these are officers! Do you get it? Officers. A man betrayed his friends, his comrades in arms. These are people who laid their entire lives on the altar of patriotism. What is it like to learn a language at native level, to give up one’s relatives, not to be able to come home to bury one’s loved...um...loved ones? Think about that for a minute! Someone has given his entire life to serve his homeland and now this brute comes along who betrays people like that. How is he going to live with that for the rest of his life? How will he look his children in the eye, the swine?! And um...Whatever went on there, whatever 30 pieces of silver those people may have gotten, they will stick in their throat, I assure you. And that um...To spend your whole life trying to keep out of view, to be unable to talk with your loved ones. That's...you know, someone who chooses such a fate will be regretting it a thousand times over.
So the quotation in the form you cite is garbled and has been interpreted in a manner which is at odds with the original context which is a specific denial that Russia assassinates traitors.
Notes on Translation
The word “загнуться” famously translated “kick the bucket” literally means “to curl up” or "to curl down". What it means here is open to interpretation. The translation “kick the bucket” can be found in Wiktionary as a possible translation of a very informal use of the word. @bashbino’s assertion that such use refers to decline and death rather than sudden death is probably correct. I suspect it is an allusion to the way plants whither and die. In 2010 the phrase was translated "they will croak all by themselves" (The Telegraph, NBC News)
In translating “прятаться” as “trying to keep out of view” I am trying to leave the question of whether the traitor is hiding from assassins or simply from people he cannot look in the eye up to the reader’s interpretation. This word can refer to social avoidance such as the behavior of a child who hides behind his mother’s skirt.
The phrase “колом станут у них в горле” refers to difficulty swallowing due to revulsion, not to airway obstruction. To make this clear I have translated it “will stick in their throat”. In 2010 it was translated "it will get stuck in their throats" (The Telegraph, NBC News)
Putin's December 2010 statement on deep-cover spies and those who betray them was not the first. On July 25, 2010 the Guardian reported a statement on the subject:
"Just imagine," he said. "First you have to master a language as if it were your own, think in that language, talk in it, [then] fulfil the task set in the interests of your motherland over many years, suffering daily dangers for you and your loved ones, who don't even know who you are or for whom you work."
Putin also said he knew the names of those who betrayed the agents. "It was the result of treason," he said, predicting a grim future for those responsible. "It always ends badly for traitors: as a rule, their end comes from drink or drugs, lying in a ditch. And for what?"
In June 2017 on Russia's TV Channel One he addressed spies currently living in deep cover abroad and again expressed his admiration for them. He also described a personal connection:
All of my work in the USSR's foreign intelligence agencies was connected not simply with foreign intelligence service but specifically with "illegal" intelligence. So I know what kind of people they are. They are special people with special qualities and special convictions, people with a particular type of personality. Not everyone can give up his present life, his loved ones, his family and leave the country for many, many years and dedicate his live to service of the Motherland. Only the chosen few can do this.