Do I use an apostrophe after the A or not?
"He got 5 As on the final exam."
I've heard that capital letters are made plural only by adding an s, but my textbook gives this example-How many A's did you write on the paper?
I'd use the apostrophe for this reason, yes.
http://grammar.ccc.commnet.edu/grammar/plurals.htm#word_as_word