This expression is not correct, because it means that the one who does that is relying upon himself first of all.
Saying, “I have done my best and I ask Allah for help” is correct. Perhaps when a person says, “I have done my best and the rest is up to Allah,” he means what I have mentioned, i.e., I have done what I can and what I cannot do is up to Allah. But the basis of this expression is wrong, rather he should say, “I have done my best and I ask Allah for help.”.