Open Source Shakespeare is a great resource for doing things like this (not my idea, I just found it). Here they've sorted on character based on number of lines. Obviously it's a little crude as Shakespeare himself shows up in the #1 spot with no plays listed, but right behind him are Falstaff and Henry V (both having appeared in numerous plays with major roles, it only makes sense), and then Hamlet (who, for only have a single play, has the most lines).
Other interesting bits...
* Othello and Iago have almost an identical number of lines.
* Anthony has more lines than Cleopatra - but he was in Julius Caesar, too.
* Behind Cleopatra, Rosalind from As You Like It has the most lines for a female.
* Romeo has substantially more lines than Juliet, though they both have quite a few. Lady Macbeth, on the other hand, has barely 1/3rd the number of her husband's lines. Tybalt barely registers with just 17 lines!