Alex: I was so angry [at/with] you. I started to looked at myself and the mistakes I'd made in my life and I realized I was overreacting.

What does he mean 'started to looked at myself''? Did he look in the mirror? Also, is there any difference between at and with.

It should be "...started to look at myself...". In this context, it means to examine your own behavior.

More often, you'll hear "angry at you". I don't see any difference in using "with".
Thanks Vorpar for the correction and explanation.