Cure? Probably no. Suppress? Definitely yes.
The term 'HIV' makes no appearance in the version of "How To Live Longer and Feel Better" I found, though he does mention promising AIDS research. I have also been unable to find any evidence of him directly saying that that vitamin-C would cure HIV, only suppress the spread of infected cells. Obviously, this doesn't mean he didn't say it, but it does mean any supposed claims may just be exaggerations or were taken out of context over the last couple of decades.
As a side note, you can't 'cure' HIV, since it's a virus and not a disease, so I assumed by 'HIV' they meant the HIV infection/AIDS.
How To Live Longer and Feel Better: Here is the text of his book, although unfortunately I this this is the first edition and not the second edition referenced in the claim. However, we can at least get a good idea of his position from this version:
The success of vitamin C in controlling other viral diseases suggests that it be tried with AIDS...
One study has been published. Cathcart (1984) examined ninety AIDS patients who had sought medical care from other physicians and who also took high doses of ascorbate on their own initiative, and he also treated twelve AIDS patients with high doses (50 to 200 g per day) of oral and intravenous ascorbate. From his limited observations he has concluded that vitamin C suppresses the symptoms of the disease and can reduce the incidence of secondary infections. It is evident that additional work along this line is needed.
Phrases like "one study", "limited observations", and "additional work" are far from seeming like he believed it was a cure. Unless a series of revolutionary studies came out between editions, it's unlikely that his position changed from "it's promising" to "it's a cure".
Vitamin C/AIDS Research: Searching through the National Center for Biotechnology Information's database of papers for papers Pauling has published that includes the word 'ascorbic', vitamin-C's more science-y name, we can see his research on the topic.
Most seem to be related to cancer, however this one is titled "Suppression of human immunodeficiency virus replication by ascorbate in chronically and acutely infected cells". It was published in 1990, about 4 years after his book was first published and definitely after the second edition mentioned in the claim.
I don't have enough biochemistry degrees to understand a good chunk of it, however I can determine that the purpose was to determine whether vitamin-C can suppress the replication of HIV by examining vitamin-C's effect on various properties of the HIV-infected cell. From the abstract:
These results indicate that ascorbate mediates an anti-HIV effect by diminishing viral protein production in infected cells and RT stability in extracellular virions.
And from the relevant section of the results:
In the controls, RT titer manifested a peak of virus production on day 4. In contrast, ascorbate treated cultures showed a striking inhibition of RT production. The first noticeable drop (64% inhibition) in RT titer occurred on day 2 with ascorbate at 50 µg/ml, followed by a progressive decline in a dose-responsive manner. Further decreases in RT level were seen with increase in both ascorbate concentration and time of exposure. On day 4, >99% inhibition in RT titer was seen with ascorbate at 150 µg/ml.
Essentially, this is saying that at 50 micrograms/ml of vitamin-C, the growth of reverse transcriptase(RT), an enzyme needed for HIV replication, began to decrease in comparison to the control, and at 150 micrograms/ml the growth was essentially stopped. Do note that this claims that the amount was still increasing (at a slower and slower rate), but not actually destroying these enzymes.
Therefore, in this paper at least, he is claiming that vitamin-C may be able to slow or even stop the spread of HIV infected cells in a patient. However, he is not saying that the cells would be destroyed, so it is not a 'cure' but a treatment.