Hmm, well, of course there are exceptions, but you generally use "what" if you're asking a question: "What do you like?" vs. "I like that." It can also be used as the subject of a sentence for emphasis: "What I like about you is..." In this case "what" is interchangeable with "the thing," and I think that's generally the case in these kinds of sentences. You CAN'T use "what" to modify another noun, though, as in the sentences I corrected. If you say "something that I like" or "something that I hate," "that" is referring directly back to "something." "What" doesn't function in this way. Basically, if it's directly following a noun, it has to be "that" or "which"--"something which I like" would also be okay--but not "what."

Of course, this is all based on American English. I think there are a couple British dialects in which "something what I like" may be okay, but in general it sounds strange to English speakers.

I hope that wasn't too confusing! It's hard to explain it well; I can only tell you whether a given example sounds natural or not.