In the game you don't find a lot out about Frank.

I can't recall the exact details, it's been a long ass time, but I think at some point you end up finding his hanging corpse. There's a note and basically implies he committed suicide because he got bit trying to get away from bill, who was being a dick.

It's real dark and sad.

In the game Bill never overcame his paranoia and was unwilling to accept Frank's attempts to let others into their lives, causing Frank to leave. Frank committed suicide after getting bit and Joel discovers his body and a letter he wrote for Bill, and later gives it to Bill as he leaves. I think it's implied that Bill also commits suicide after Joel leaves.