To what extent is the phrase "I did my best, and the rest is up to Allah" correct?

This statement is incorrect because it implies that the doer relied on himself first. The correct thing to say is: "I have exerted my effort and I ask Allah for assistance." Even though the speaker might mean by this statement, "What I am capable of, I have done, and what I am not capable of, is up to Allah," the original phrasing of the expression is wrong.