help with power supply

After a long search, I finally found a suitcase.  It isn't exactly what I was looking for, but I want to get started on a build, so it'll do.

I have a lepai LP-2020 amp and I'm going to use a small 12v battery, but I don't know what I need besides the power supply (I'm looking at this one, but I'm not sure what else I need).  What parts do I need to order to be able to run on battery when I'm out and about, and run from a wall outlet when that's available.  I'm not sure if I'm making it more difficult than it should be, but I'm worried if its plugged in, it will overcharge the battery, but I want the battery to charge when plugged in, too.

Any help will be much appreciated!  I love looking at others' build pics so I promise to post mine up when I get it built.

Here's the amp, it came with a cord that reads 13.5v 3.0 amp. Could I get by with that or is that .3 volts going to be a problem? Maybe it would be easier to use that cord for the wall and a regular car battery charger to recharge?

I'm planning on picking up something along the lines of this.  If I go the car charger route, I would just take the battery out to charge, I have a big car charger.  I was just hoping to keep everything internal, but I didn't know what all it entails.

how about a 13v dc supply wired in parallel to the battery, with a diode to prevent back current when a fully charged battery gets connected.
after diode volt drop it will leave something like 12.5v, enough to keep battery topped up  but not enough to overcharge it...
battery can take up a bit of the load if power supply will be not enough for the amp.
got a plan like that for my system, i want to get a small suitcase generator for a back-up, when battery levels go too low.

Use a Schottky, eg:  http://es.rs-online.com/web/p/diodos-schottky-y-rectificadores/4860216/

crossed wrote: Since the Lepai amp comes with a 12v AC adapter will 13.8v harm the amp?

The Lepai amp (assuming its the Tripath 2020 one) will be fine with 13.8V - but don't go any higher.
I'd suggest measuring that voltage though before plugging it in - if its a switch-mode supply it'll be fine, but if a transformer one, the open circuit voltage could be a good deal higher.

Also - I'd suggest :-

Unless you can design, build, and test a circuit for dual battery charging and powering the amp ..... well just settle for AC operation via your AC adapter OR battery powering - and leave out the simultaneous operation.... also use a "proper" battery charger for the battery - anything else will likely not charge the battery properly and/or shorten your batterys life.