Clearly he believed he was a genius. And while this is perhaps slightly more controversial, it also seems to me to have been obviously true. But of course we all know from Plato that true belief is not enough for knowledge; though it is controversial what exactly are to count as good reasons in general, it is almost universally held that one cannot know on the basis of believing for bad reasons. So were Nietzsche's reasons good? Was it even possible for his reasons to have been good? What could be good reasons for that sort of thing?
It's almost a stereotype that geniuses are misunderstood and neglected in their own times. But the stereotype seems to have only a shaky basis in reality; quite a lot of revolutionary thinkers were wildly controversial in their own time, but they were of course centers of storms of controversy because they were also targets of enormous amounts of attention, because they were widely considered extremely important. Those who totally escaped notice among their contemporaries to be appreciated only later seem to be pretty unusual. On the other hand, those who escape notice by anyone ever because they're just totally mediocre are, of course, extremely common. So Nietzsche shouldn't have concluded he was a genius just because he was misunderstood and under-appreciated, and I don't think it's reasonable to attribute that theory to him, either.
Admittedly, being widely acclaimed in your own time is certainly not proof of genius, as there have been plenty of widely acclaimed cranks, and there have certainly been some under-appreciated geniuses, so perhaps Nietzsche should not have worried too much about not having widespread acclaim; perhaps one shouldn't appeal to that either way. But what other evidence could their be?
Those close to someone are likely to be biased in their favor. If they care about the person for other reasons, they're unlikely to be too critical of things that are important to the person they care about; indeed, they may genuinely value those things more than they otherwise would just because they associate the things with their author. Also, people who share similar views are more likely to become close, so anybody who becomes close to you is likely to think you're right about more things than you are, because they're likely to be wrong about some of the same things that you are (and so think you're right about those things). So while Nietzsche had some friends who thought fairly highly of him, it is unclear how much he could get from that.
As a student, he was hailed as a brilliant classical philologist. However, he never did very much work in the field, so it is unclear whether he had sufficient grounds for even believing he was a genius in that area; some are far better at impressing teachers than doing independent work, so it is risky to draw conclusions from the evaluations of teachers. And in any event, Nietzsche clearly thought he was a brilliant philosopher, not just a brilliant philologist.
So what's left? He came up with results that seemed right to him? But who isn't able to manage that?