Is it an Essex dialect or something else?

Being not a native English-speaker and not familiar with the dialects of the English language, especially the British ones, I'm wondering about an interesting fact for a long time. A British music group, The Prodigy, has released a single Breathe, and there is a lyrics line in the song: "Come play my game". Knowing the English language, I'd read this as [kʌm pleɪ maɪ ɡeɪm], but it's pronounced by the vocalist like [kɒm plaɪ mɔɪ gaɪm] (or something very similar). Another their single Firestarter features lyrics "I'm the pain you tasted" is sang like [aɪm ðə paɪn jə taɪstɪd].

I know that the band is from Essex and the Essex dialect exists, but I really couldn't find much information on it and I'm not really sure if it's true. I've also came across a mention that it also might be relevant to "the Southern English dialect spoken by chavs in London" (found at a non-English lingvo-forum), and I don't know if it's true as well.

What dialect such pronunciation is characteristic of?

I'm not a linguist, but I come from the South East of England. This way of talking is now commonly described as Estuary English.

There is a difference between dialect and accent. Dialects have different words for the same concepts, and sometimes different grammatical structures from each other. For instance the speakers of US dialects (there are several) talk about the 'hood' and 'fenders' of a vehicle while the speakers of British dialects (including Standard English) talk about the 'bonnet' and the 'bumpers'. Similarly many Americans will use 'gotten' as the past tense of 'to get', whereas it hasn't been used in British English for a very long time.

Accents, however, are different ways of pronouncing words. It is perfectly possible to use Standard English (whatever that means) but to speak in a regional accent, people do it all the time. For instance most british politicians speak the same dialect but have many different accents. What the OP is talking about is an accent. Whether that is an Estuary English accent, an Essex accent (and if so which one) or a 'Revolutionary Punk' (John Lydon?) accent I leave others to judge.