Agreed that nothing exotic is necessary to get to a grand. A .308 with the right load gets you there accurately and cheaply. The cooler stuff mentioned by SAF (.260 Rem, etc.) will get you there with less influence from the wind.
If I had $5k to spend on a rifle, I'd probably start with a better action like a Surgeon, pick a barrel you can actually get in 6 months or less, and have at it.
Stock selection can make a huge difference in how well you shoot a particular rifle, so I'd recommend you get behind as many different types as you can before making any decisions.
Go over to the Hide and read for a couple of days and you'll get yourself all up to speed on what those guys think is working.