It is possible to make DIY a similar laser. The laser useful for this purpose is Q-switched Nd:YAG laser at high repetition rates. I've made a similar module but not q-switched and at low repetion rates for about 90€ .
These lasers works because they burn only the dark part of the skin without pain in a very short time(3*10^-9 s ), and lately the human blood carries out the rest of the damaged-->tattoed part of the skin previously hitted by the laser.
The problem is that the laser must be high powered, and/or focalised in a small spot, and if the tattoo is big this is a problem!
Then if you are good with physics and laser , than try to build yourself this laser. You can find useful information on the web and bought laser elements on ebay at reasonable prices.