Hi,
An arduino can work stand-alone if you adjust the code. If you like to and already are going to use it in your PC, you could control it with your PC as well.
Powering by USB may be possible, depending on the number of leds, current flowing and type of strip.
Since USB only provides for 5volt and 500 mA it probably isn't a good idea though. Each Led can draw up to 20 mA, there are 3 leds in a single RGB-led and your USB port will probably start to protest if you connect more as 8 RGB-leds.
If you're connecting your project using a non-powered USB-hub, you'll need to take care other devices won't draw much current as well. Besides, you'll have to find a 5v strip while it probably is easier/cheaper to find a 12 volt one.
But why not use 12 volt ?
If you're using a desktop you already have a 12 volt power supply capable of working with much higher currents.
Personally I just cut off one of the older 4-wire Molex-connectors (Yellow=12v, black=GND, black=GND, Red=5 volt), normally used to power internal peripherals and use that to power projects.
There still will be a limit to the number of leds you can connect this way, each supply has its limits, but you probably can turn your PC into a Christmas tree and light a complete room.