Ask yourself why you want to get a mining rig? For fun? A hobby? Or to become rich? If it is for fun or hobby, then great! If it is to become rich, forget it!
You can get any cheap computer, cheap CPU/RAM. Don't need a big HDD or SDD. Most important is the GPU and a big enough power supply unit to run it. Definitely get AMD GPU, not Nvidia for mining.
As others said, go here and pick a GPU, but most people think a 7950 is a great choice for price/watt and price/hash ratio:
https://en.bitcoin.it/wiki/Mining_hardware_comparisonNow take the hash rate (NOT MHz speed) of the GPU you picked and put into calculator:
http://www.bitcoinx.com/profit/For a 7950 with 550 MH/s, you will get about 0.0247 BTC/day at current network difficulty level. Keep in mind that as the network's whole hash rate increases, the difficulty will also increase. Meaning you will earn less and less BTC over time.
Good advice. Let's say you buy a cheap computer with a 7950. It will cost you $600 or so ($300 for the 7950).
If you mine 0.0247 BTC/day, you will make about $3 per day at the current price, but don't forget about electricity. Electricity rates vary, but let's say electricity costs $0.75/day. The net is then $2.25/day.
At $2.25/day, you will pay for your mining equipment and start making a profit in about 9 months.