You can get the Lepai 2 channel amp for like $20 on Amazon to power the car speakers. But personally I don't like the look of car speakers on an arcade cab.
I use a router to route out my speaker "grills" and place the speakers behind them. I have used the Logitech 2.1 setup found here:
http://www.amazon.com/Logitech-LS21-Stereo-Speaker-System/dp/B0015C30J0/ref=sr_1_3?ie=UTF8&qid=1408399753&sr=8-3&keywords=logitech+2.1I put the speakers behind the grill and the subwoofer down in the bottom of the cabinet and it is more than capable of hurting your ears if you crank it up. I usually set windows to about 3/4 volume, and the speakers to about the same (it has a remote volume control) and it is perfect for most games. I route the grills above the monitor so the sound is focused down on you and not spreading throughout the room as much.
Here is a quick photo rundown of what I do:
First I draw a circle where I want the speaker holes to be, then I use a straight edge guide with the router to route 1/4" wide slots. I keep the slots within the circle I drew to give it a shape, but you could do just a square too. (You could also use pegboard as a drilling template and just drill out the holes within a circle or whatever shape you want)
I then paint it from both sides at all different angles so the insides of the slots are blacked out. This helps with the painting later so you aren't getting drips trying to get enough of your finish color inside the slots.

Then since I use those Logitech speakers, I cut blocks that fit tightly in the squares below the speakers, line up the speakers and glue and nail the blocks into place. There is a slight taper on the blocks because the speakers are slightly tapered.



I then run one small screw through the bottom of the speakers securing them to the blocks. The wires always end up just barely long enough to connect to the subwoofer, which houses the amplifier, but if I needed extra length, a regular 3.5mm extension would work just fine.