There is no man hath a virtue that he hath not a glimpse of, nor any man an attaint but he carries some stain of it.
You have to consult the context to understand what is being said here. The two hes in this sentence do not refer to the preceding no man and any man but to the topic of Alexander's speech, the very complex Greek hero Ajax:
This man, lady, hath robbed many beasts of their particular additions; he is as valiant as the lion, churlish as the bear, slow as the elephant: a man into whom nature hath so crowded humours that his valour is crushed into folly, his folly sauced with discretion: there is no man hath a virtue that he hath not a glimpse of, nor any man an attaint but he carries some stain of it ...
The part you ask about may be paraphrased:
There is no virtue, in any man, that Ajax does not have at least a hint of; and there is no vice, in any man, that Ajax is not to some extent tainted with.