Praise be to Allah.
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.”.