You can say certain phrases in English, but that doesn't mean they make sense. In your context "can't resist [something]" implies that the person must give in to some temptation or compulsion, even though they otherwise might want to.
Although he's not a regular smoker, he can't resist having a smoke if he sees a friend light up nearby.
I'm on a diet, but I can't resist that delicious-looking chocolate cake.
She hates to be wrong and can't resist getting in the last word of an argument.
Based on this usage, "he can't resist being the best" is a little weird. Grammatically it's fine, but it begs the question why he would want to resist in the first place.
Instead, I would say something like:
He was always compelled to be the best.
He couldn't stand to be anything less than the best.
No matter what the activity, he was always driven to be the best.
and various others.
Note (as the other answer points out) your second example is a double negative -- which is not itself wrong, but you do have to be careful that the sentence makes logical sense and means what you want it to mean. Replace "can't resist" with "must" (which retains much of the overall meaning) and you get:
He [must] not be the best of them all.
This is the opposite of the first example, and also doesn't really make sense. Why would he be tempted not to be the best?