A similar claim is often made, but limited to the use of marijuana. For that, the Washington Post has a good overview. The results for overall drug use are similar to those results.
Drug use by race/ethnicity
The dataset that is used most often to evaluate claims like this is the National Survey on Drug Use and Health.
You can view the data here for 2011.
You can view a table showing data from 2012 here.
Basically, while white people tend to try drugs in higher numbers, looking at the "past month" value, it can be seen that black people do drugs in slightly higher numbers.
The same can be seen when looking at the number of days in the last month that Marijuana was consumed (the graph is generated from the 2011 data set):
Since some are cropped, the data labels are: (1) White, (2) Black/African American, (3) Native American/Alaskan Native, (4) Native Hawaiian/Other Pacific Islander, (5) Asian, (6) More than one race, (7) Hispanic.
This is a trend that can be seen over the years:
Arrests for drug offenses by race
slate.com and politifact.com have created a graph from data from the BJS:
[T]he National Research Council report says, "In recent years, drug-related arrest rates for blacks have been three to four times higher than those for whites. In the late 1980s, the rates were six times higher for blacks than for whites."
Human Rights Watch says:
In every year from 1980 to 2007, blacks were arrested nationwide on drug charges at rates relative to population that were 2.8 to 5.5 times higher than white arrest rates.
Disparity between drug use and drug arrests
The above data shows that while black people do marginally more drugs, they are arrested disproportionately more often than white people.
politifact looked at a similar claim and concludes:
[...] African-Americans don't use drugs at a higher level than whites but wind up going to prison six times more. [...] We rate his claim Mostly True.