This route is accessable from several different trailheads in the Cooke City area. Probably the easiest route to find would be via the Clay Butte trailhead-- after Green Lake, a social trail takes off up the beautiful Sierra Cr. drainage, which ends up at Forsaken Lake after ~20 miles of approach.
A more scenic and Alpine approach (but tricky to find) would be via the Island Lake trailhead... follow this trail in for about two miles, where a fainter trail will branch off to the right. Follow this to the East of Lonesome Pk, where you will pass by Albino Lake and Jasper Lake. From there, you will cut cross-country (it's all well above treeline) to the NNW past two lakes that either go by Maryott and Donelson lakes, or Felis and Lynx lakes, depending on which map edition you have. You will then drop down to Crystal Lk and down into the Sierra Cr. Drainage, which you will follow to Forsaken Lk.
Once Forsaken Lake is reached, follow the Eastern shoreline, and make for the mountain furthest to the right, which kind of resembles a giant armchair in the sky. (note: my photo "Forsaken Lake, 7 AM does NOT show Castle Mtn! This is Castle Spire, the next peak to the left.) Take the path of least resistance into the high cirque on the peak. You can then either cut right up the cliffs (a respectable scramble), keep going up the frighteningly loose boulderfield (I know of someone who was killed when one of these boulders rolled on her), or take the prominent, steep snowfield. Whatever the case, it's a phenomenal (and pretty easy) alpine climb--probably my favorite mountain ever.