[I was typing this while Bob was posting, so there's some overlap...]
You might also use any unwanted PC power supply you have knocking about. Recent ones produce high current at 12V, for powering the motherboard. It needs a bit of soldering, but it's not complicated (basically many more wires than you will need).
Older simple chargers should work fine, but the voltage can go quite high, but again a car radio should be fine with that. Modern chip-ccontrolled ones will try to charge the radio, which isn't helpful. If they don't see the load (the radio) behaving as a lead acid battery they may shut down.
OTOH, if you wire up charger -> battery -> radio, it should be fine, although you _might_ get radio interference from the charger (unlikely, but interference suppression won't be a primary objective of the charger designers!). Car radios usually have suppression built into their internal wiring, so again you might notice nothing.
The only nuisance I can think of is earthing (grounding). Car radios use the electrostatic vector of radio waves: being inside a big steel box means magnetic aerials (as found in traditional transistor radios) don't work.
To get the best reception, you need a good ground connection through from the case of the radio to the aerial base. Fitting car aerials to tin garage roofs works well, as long as you use a co-ax aerial cable through to the aerial socket into the radio, not a bit of string. If you use a PC power supply, you might need to _insulate_ the aerial mounting instead, as the power supply needs its earth, and there should only be one earth point in this context. If you don't insulate it, you might find the RCD trips or you get corrosion on the roof around the aerial mounting.