Attach the 1/8-inch end of each RCA cable into the "line out" jack of each computer. If your computer doesn't have a specific input for line out, then the headphone jack will work as well.
Attach the red and white (left and right) stereo plugs, on the other end of each cable, into the two sets of RCA inputs on the DJ mixer. These are the same sets of inputs used for CD players and/or turntables, but will be used here for the two computers.
Sign on to YouTube on each computer. On each YouTube screen, create a video playlist of the songs you want to play. When you go from one song to another, you will be alternating between computers, just as you would between turntables, so keep that in mind when arranging each playlist. Make sure the volume of each computer and each YouTube screen is at least 3/4 of the way up.
Press "Play" on the first YouTube screen. The audio will be going to input 1 on the DJ mixer. You can adjust the volume with the up/down input 1 fader on the mixer. Make sure the cross fader, in between inputs 1 and 2 on the mixer, is panned all the way to input 1.
As the first song nears an end, press "Play" on the second YouTube screen. Make sure the input 2 fader on the DJ mixer is raised to zero. Gently slide the cross fader over to input 2, and the first song will fade out as the second song fades in.
Repeat this back and forth between inputs/YouTube screens as much as you like.