In formal, correct grammar, "at" would not be correct. A grammatically correct version might be "I'm dying because of this" or "I'm dying from this," but either one would sound strange in the context you have described. Grammatical or not, "I'm dying at [something funny]" has become a common idiom, especially on the internet, and it is not readily replaced by a truly grammatical alternative.
If you want to express the same sentiment but are determined to be grammatically correct, you could say "I'm going to die laughing at this!" or "That's so funny I'm going to die!" It's easy to get wrong, though, and just slightly misusing any idiom involving death could make people uncomfortable.