If there is no self how can Nirvana be described as bliss? I've heard Nirvana described as a state of bliss, but if there is no self then how can it be described as bliss when nobody is there to experience it?
According to Buddhism, even when I am typing these words and seeing the screen, there is no one there. The 'person' isn't an existing thing, it is just a label we put on things to describe how things work, and the label of person refers to a large group of many different mental and physical processes. 

If there is ultimately no self, does that mean that all experience is impossible? No. There are individual parts that preform functions and made experience possible. When I look at the computer screen, it isn't some sort of seperate "me" inside my head that is seeing the screen. Rather, the coming together of the object, my eye, and and my brain, work together and as a result, awareness of the screen arises.

It's the same way with Nibbana. There isn't anything there that you can really call a person, but there is Nibbana, and there is consciousness that is taking Nibbana as an object (called Lokuttara-Citta in the books of Abhidhamma). It is this consciousness that is blissful.