The
Gleaves class was a variant of the
Benson/Livermore class, I believe. Skywave makes some nice 1/700 kits of vessels in that class.
John's right, though: the "actual"
Caine (i.e., the one in the novel) was an old four-stack destroyer - heavily modified. That's one of several discrepancies between the book and the movie.
This is, as it happens, a subject with which I'm familiar. I assign the novel in one of my courses every year, and students sometimes (not often; I warn them) try to get out of doing the assignment by renting the movie. It's a pretty good flick in many ways, but in my opinion a pale shadow of the book. One other major discrpancy: Bogart was ludicrously old for the part of Commander Queeg. The character in the book graduated from Annapolis in 1937, so he was about 28 years old at the time of the "mutiny."
Some time back I encountered an interesting article about the book in the
Naval Institute Proceedings. The author of that article (I can't recall the issue) noted that Herman Wouk had served on board the U.S.S.
Zane (DMS 14), which, like the fictitious
Caine, was an old four-stacker with one stack removed. That ship would be a good place to seek inspiration for a model. There are several photos of her at the Navsource site.
So far as I know there have been two plastic four-stack destroyer kits, both of them now quite old: the Airfix one on 1/600 scale (issued as H.M.S.
Campbeltown) and the classic Revell one on 1/240 scale. According to the bible on the subject, Thomas Grahan's
Remembering Revell Model Kits, it was originally issued in 1960 as the U.S.S.
Buchannan, then reissued under the name U.S.S.
Aaron Ward (in 1962) and twice as H.M.S.
Campbeltown (in 1972 and 1979). The Airfix kit is mighty basic, but could be made into a nice model with some work. The Revell one is, for its age, pretty good. It does suffer from one awful blunder in terms of accuracy: somebody put in a great deal of effort indicating wood planks on the main deck. (Those ships had steel decks.)
I'm sure there's at least one four-stacker kit in resin. And if you're really ambitious, there's the Bluejacket wood kit in 1/192 scale.
Hope this helps a little. Good luck.