Isolate your thesis, the single concept that you mean to express if nothing else. If necessary, scrawl it on a piece or several pieces of paper and stick them to the wall above your work area. Ask each other point in the paper, "is this relevant to what I'm trying to say? Does it advance my argument or confuse things?" Cut anything that doesn't get two positive answers.
Check the flow of evidence and argument. Keep your evidence sequential. However you organize the paper, make sure there is a rational sequence of evidence that a reader can follow. Each piece of evidence should build on the ideas that have come before, and general points should be gradually "unpacked" to reveal their details.
Go all the way through the paper again to eliminate verbosity. Flowery phrases and extravagant erudition gather fog around your thesis. Especially if writing for a university class, whose professor has likely read it all, be as terse as possible. Concern yourself with "tightening" up the writing so the same information is conveyed with fewer words.
Remove all excessive uses of technical vocabulary and all cliches. Every academic discipline has its buzzwords, assumptions, and given concepts, but relying on them for an audience that already knows them too well is only boring. Labeling is not the same thing as understanding. For example, if writing on Gogol, avoid standard characterizations such as "realism," "romantic," or "grotesque." These words are part of the literature on Gogol and many other writers and show up constantly.
Check the paper finally for small errors. Go back through the paper more than once. As you check a second or third time, start to look for smaller errors such as the improper use of semicolons, abuse of the words "therefore" and "however," double negatives, inconsistent citation methods and other smaller variables. Always double-check your bibliography or footnotes for consistency, and check the spelling of every proper name, even your own. Do not rely on spell-check programs, which can tell only that a collection of letters makes a word, not that it makes the word you intended to use.