How to battery power?

Nick,
You won't use a car battery for this at all.
You will need an ac to dc converter.The converter will
supply your power.It will have a red(hot)and black(ground)
set of terminals for the amp.Radio schack has a 20 amp
product that does not use a fan and costs about $40.00.
If you need more juice(I doubt it)...I am using a 52 amp
product by Pyramid(it has a fan,but I am running two
powerful ADS amps off it,and needed more power).
To power your amp,you will simply plug the converter into
the wall,flick the switch on the converter...and you're
up and running.It works great!

Your plan for the car amp sounds workable, aside from having a car battery in your listening room. However, if you were to take the cost of the car amp plus about $100 for battery and charger, you might then post that total amount as your budget and ask for amplifier recommendations at that price point.
It is likely that your member friends can suggest some options that did not occur to you.
Here's one now.... An Onix SP-3 can be had used on this site for $500. It is a tubed integrated with 38 watts per channel output and very sturdy construction. It has two inputs which is one more than you get with a car amp and it comes dangerously close to the big boys sound wise. Check it out.

You need a high capacity battery that can deep cycle which will cost over $100, probably closer to $150. You need a smart charger that can handle the battery properly which will run you close to $100.

I agree with macrojack that with this plus the amps you can get something to plug in the wall just as cheap or cheaper with a lot less hassle.

How much is the amp?