It might push your £500 limit, but I would look to go Genesis profile amp, DLS comps up front, and a twin voice coil JL 10" sub (boxed) to load the amp at 2 ohms (lots of clean juicy power).
If you must run rear fill just some co-ax's of a similarly decent brand, head unit powered, would be more than sufficient, nothing more.
As B-Chi says, decent gear installed badly will sound poop, so don't skimp on cabling and invest time and affort in routing cables properly and making decent connections.
Definitely don't skip adding additional sound deadening to the doors, can be worth up to 2 or 3Db.
Long term, consider changing your head unit for something that will spew out a nicer signal. Nothing wrong with the JVC per se, but once you start running clean amplifiers and speakers it will highlight any deficiencies in the signal they're getting....