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.
One must be very careful with the idea that there is no self. This is not quite what the early Buddhist texts say. What they say is that when one examines one's experience (the five khandhas: form, sensations, perceptions, volitions and cognitions) one does not find a self, nor anything that belongs to a self (Alagaddupama Sutta, Majjhima Nikāya, Sutta no.22).
This is because experience is constantly changing, whereas by the definitions of the time a self is not changing. The unchanging ātman which is not affected by actions or events is a feature of the early Upaniṣads and the Bhagavadgīta. But if your experience is always changing, which can be confirmed by simple introspection, then no unchanging thing will ever be perceptible or comprehensible. At no point did the Buddha say "there is no self". He strongly implies that there is no possibility of an unchanging self.
Thus if there is any unchanging element to a human being we cannot experience it. And thus from a Buddhist point of view it is irrelevant. In the Sabba Sutta (Saṃyutta Nikāya, 35.23), the Buddha defines "everything" (sabba) as the āyatanas: the six kinds of sense object and the corresponding kinds of sense faculty. Everything of interest to the Buddha is found in experience. And really this makes sense because outside of experience can only be speculated about (there are philosophical problems with this view, but it is conventional enough that most people accept it without question).
Where Descarte said "I think therefore I am", the Buddha might have said "There is experience". He does not conclude that this implies "being", he just acknowledges that there is experience.
On the other hand we do experience ourselves as having a first-person perspective. I experience myself as a self. The Buddha seems to have acknowledged this, but pointed to the fact that feeling like we are someone is just an experience and as such marked by the same characteristics as all experiences: impermanence, disappointment, and insubstantiality.
The problem from a Buddhist point of view is that we are intoxicated with sense experience. And we have a view that happiness is about maximising pleasant sense experiences and minimising unpleasant ones. So we chase experiences in pursuit of happiness. But the nature of experience is impermanence and thus they cannot satisfy.
In the same way, just because there is no permanent feature of experience does not deny that there is an experience we can call "bliss". It is no different from other experiences. Even nibbāna is an experience. It's just that nibbāna radically transforms how we perceive other experiences once we have it. After Nibbāna we stop thinking about sense experience as leading to happiness. It's just what it is and nothing more. Happiness is not something that comes from sense experience at all. It comes from not being intoxicated with sense experience.
So in this context the idea that nibbāna might be accompanied by an experience of bliss is certainly plausible, but not very interesting. In fact as with the Dhammapada verse mentioned above, it's doubtful that sukha even means "bliss" in this context. In fact it is more likely to mean "well-being" or even "happiness".
To make answer clear to you I describe difference between the self, soul, nibbana and parinibbana. There is no soul but there is a collection of temporary things which can call as self. Self is consists of body and mind. Anyone can easily understand that body is not immortal but it is very hard to realize that mind is dying and generating in ultra high speeds. One particle of the mind is called as naama and you can understand the structure of naama in following image which I used to describe same in previous answer.
here 3t1 is the time taken to one naama's existence. naama has 3 states in it's life time - Borning, Existence and destroying. There are billions and billions of naamas generated in eye blink time. So that's why we feel continuous mind. So collection of this continuously dying mind and body you call it as self.
After Intelligent person realizing that there is nothing which can be called as me and mine, He realizes the uselessness of giving comfort to the body to generate some collections of naamas to make it feel as comfort. Due to the efforts of giving comfort to the body with foolishness, we were born in the hell in very very big numbers compared to the heavens and Brahmans. So Intelligent person determine to stop generating these naama and meditate more. After achieve nibbana person get ability to stop naama after parinibbana.
The Bliss can be experienced by the person who attained nibbana and not yet had parinibbana. He knows the danger he has bypassed with the dhyanas and that will make blissed happiness in the mind. After parinibbana there is no mind, so no physical body and self. Blissness after the parinibbana is there is no self to get hurt by the hell and sadness including upper states.
Nibbana is experienced by the mind. It feels good to the mind. No 'self' is required to feel Nirvana. An analogy is giving a depressed person an anti-depressant drug. The anti-depressant drug is stimulating pleasurable feelings in the nervous system (rather than to a 'self'). Similarly, Nirvana is felt or experienced by the nervous system to be blissful. No self is required to know Nibbana.