Your budget is not consistent with your goals, if I remember the spec for Call of Duty, Counter Strike, and similar games. Not familiar with Rakion.
To download movies, you need disk space. Gigabytes at a pop.
To view those movies, you need a pretty good video card, which could either be AGP or PCI based. I've viewed movies on an AGP 8X card and it isn't always a pretty experience. Nor is it always bad. PCI cards can go faster but they take up a PCI slot, which is a limited resource.
To download music, you need less disk space than your movies will use, but you need a decent sound card.
You also need not less than 2 Gb of RAM (and 3 Gb wouldn't hurt).
Alienware isn't necessarily a waste of money, but if you have no desire to get one, that is surely your call. My problem with all of this is that the last time I priced boxes with good gaming graphics, good audio, and enough disk to be useful for holding several games without having to remove and re-install frequently, your stated budget would require you to make compromises.
I'm going to make a suggestion, but it requires that you have access to one of these places: If your parents have a membership in one of the many discount merchandise chains, take a look at their offerings. Typically they buy in bulk so get some pretty good deals on machines that, bought individually, would be way out of your price range. The machines they buy like that also tend to be somewhat over-configured in terms of peripherals, perhaps a little hotter than they might otherwise have been.
I'll be specific: At least four years ago I went to Sam's Club and got an HP Pavilion with 2.8 GHz dual-core Pentium IV CPU, really great RAM, good (and fast) disk space, a nice nVidia graphics card and a Yamaha sound card with full midi abilities. Plus several USB slots, a multi-slot camera memory card adapter for the seven most common memory cards, plus serial and parallel slots, a fire-wire port, two DVD readers (one of which is also a writer), and a flat-panel 17" LCD monitor. For less than $1700 total system price, which at the time was a true steal. Wife's was a similar model but 3.2 GHz dual core, 19" flat panel, larger hard drive, and an ATI card for graphics, for just a couple of hundred bucks more than that. With a different disk adapter and a little more memory, either of those could have been small-business server machines.
If you can get to one of the discount merchandisers, you should be able to get an equivalent but updated deal on a smaller machine.
If that isn't to your liking, watch for liquidation sales. It is possible to get "last year's model" of some nice system for 40% off and still get a new-system warranty on it. I bought our laptop that way. Brand new out of the box, ready to be registered for the first time, fully loaded, and with all software. It also passed the "GMA" test when I visited the Windows Update site, so it was a legit copy of WinXP.
If that STILL doesn't get you going, invest in a copy of the magazine called "Computer Shopper" and thumb through that to get some ideas. You might see something that tweaks your interest.