My recommendation would probably be the Asus G73 laptop. It may not end up being as powerful as something you build yourself, but I've found it to be able to run most modern games at maximum or near-maximum settings quite well. Besides, it's a laptop (and therefore portable). I've been using it for the last several months and have been quite liking it. In terms of budget, I don't remember the exact price I got, but it was somewhere around $1,000 USD.
Building your own will grant you the most flexibility in terms of cost/power, but I'd highly recommend this laptop if you don't want to put in the effort.