The direct route is the Bucklin Trail. Take Rte 4 west over the pass and down past Pico, past the AT parking area. Just before a bridge will be Wheelerville Road on your left, with a large number of mailboxes at the end of the dirt road.
Go several miles down Wheelerville, I forget exactly how much, but you will come to a point where the road takes a sharp turn to the right, there's a gated forest road ahead of you, and a parking lot on the left which is the start of the Bucklin Trail.
The trail is very moderate, no crossings of issue, and will hit the AT/LT at Cooper Lodge. I believe it's yellow-blazed. From there, you can take a blue-blazed spur trail up a short, steep distance to the summit of Killington.
Leave the summit through the trees on the walking trail and come out at the top of the gondola. Walk down a bit and you can stop in the summit building for a rest stop and food. From there, come back the way you came, or, if you're feeling adventurous, start walking down Great Northern, following it to, I think High Traverse? In any case, keep the summit lodge on your left as you curve around; you'll pass the top of the North Ridge Triple (the Rime and Reason trails) on your right. As you approach the woods, you'll see a trail continuing past a number of warning signs telling you that you're screwed for leaving the ski area boundary. Keep going, and in surprisingly little time you'll be back at Cooper Lodge.
If you want to have more fun, you can head up that forest road and visit Mendon (a NE100 peak) then over to Killington. The road has some unmarked turns and the peak is a bushwhack and if you want to go that way you'll need more information.
If you want more fun without the difficulty, leave a car at either the top of the pass on Rte 4 (across from the Inn at Long Trail) or at the AT-signed parking lot a few miles west on 4, and from Cooper Lodge follow the LT/AT north towards Pico. Stay left on the LT/AT for the parking lot, bear right on the Sherburne Pass Trail, make the side trip up to the top of Pico (a NE100 peak), then continue to the top of the pass.