Does Kousei end up with Kaori?

No, Kousei doesn’t end up with Tsubaki. In the anime, Kaori dies due to an unknown illness, thus leaving Kousei to Tsubaki. But, Kaori does confess her love for Kousei in a letter.

Kaori’s letter

In the letter, Kaori apologizes for all the things she made Kousei do. For all the pranks she played, and even goes on to say that she doesn’t actually like Watari. She lied to everyone about Watari.

The only reason she came for Watari was to get close to Kousei, and she even asks Kousei to apologize for the same.

What happened to Kousei after Kaori died?

Kousei follows Kaori’s advise and continues his progression to being a great musician. He plays with his heart, and not out of spite. Which makes him all the more better.