It's fairly simple. Look up some system requirements for the games you know you want to try on PC. Make a note of CPU, RAM, and Video requirements.
CPU and RAM are the easy parts. For Arkham Asylum: Buy or build a computer with a 2.5 GHz dual core or better CPU and 4 GB or more of RAM. That's it. You probably won't be able to buy a computer slower than that these days anyway. If you're buying a computer pre-built, brand name does not matter. Just go with whoever has the best deal.
A quad-core is always better, but don't rely on it to make up a difference in clock speed (again, not that you'll find any quads below 2.5GHz today anyway).
If you choose to build, bear this in mind: Intel is more expensive but will generally be higher performance. AMD will almost always be lower price, giving you more bang for the buck, so if you're on a tight budget, definitely go AMD.
The more complicated part is the graphics, since the number (eg "ZOMG GT 250!!") doesn't necessarily coincide with performance. You must look at benchmarks to get an idea which cards are sufficient and which aren't. Also they're different between the two major manufacturers (ATI/Nvidia).
Note: Do not get distracted by people who pimp one over the other, do not be distracted by people saying do not combine an ATI video card with an Intel CPUs or Nvida with AMD CPUs. These people are wrong.
Arkham Asylum for example has a minimum spec of Nvidia (GeForce) 8800 or ATI (Radeon HD) 3800. Those are fairly old. The GTS 250 that's in my desktop is superior to both those and only cost me $100 when I got it. Look up the fucking benchmarks. (Such as this one [http://www.videocardbenchmark.net/video_lookup.php?gpu=GeForce+GTX+460])
If you're on a budget, print out the benchmarks and have them with you when you shop, or better yet buy your video card online from NewEgg since you'll be able to look at a much wider variety of cards than any store will carry. You will find that for small increases in performance, the price jumps up dramatically. That's just the way it is with video cards.