What are you doing specifically with audio. Are you capturing one track at a time? multiple tracks in real time? short duration songs, longer pieces? Thats all going to determine what speed and what capacity you will need in your hard drives, and given that drives are still insanely expensive, thats a definite concern. If you need high speed 1-2TB drives, you probably won't be able to keep it under $1,000. If we assume you can live with a smaller, standard 7200 rpm drive then we can probably put something together for a desktop for that price. A laptop in that price range is possible (look at one of the Dell Studio 15 series for example) but isn't going to be ideal. Realistically you are going to want a very high end sound card too for your audio capture unless you already have an external interface of some sort., and even then you'll want something decent for playback, so factor in another $150 or so for a good card. That doesn't leave you much room for a decent gaming video card at all.
Basically your low to mid range desktop gaming computer cost is going to look something like this:
Power supply $70
Operating system $190
Which would put you right at about $1,000. You could skimp on the case a bit and get a crappy one, same with the processor, and then you might be able to come in closer to $900, but I wouldn't reccommend it. A good mid to high end computer is going to start closer to $1,500 though.
If you want, I can write up a $1,000 build, but it won't do what you are looking to do with it well. Neither will the sub $1,000 laptops. A good rule to remember with laptops is 25% more expensive for %25 less performance over a comparable desktop.