Drive to Durango and follow signs to the Durango and Silverton Narrow Gauge Railroad. It's near McDonald's and has a large parking area nearby. Buy a ticket for the train that stops at Needleton and ride the train 2.5 hours (~30 miles) to the Needleton stop. The train will drop you off next to a suspension bridge that crosses the Animas River. From here, it's a 6 mile hike to reach Chicago Basin. Note: you can also take the train from Silverton and be dropped off at Needleton.
Taken from Windom Peak, Photo #1 shows the standard routes up Mt. Eolus and North Eolus. To reach North Eolus, follow the Mt. Eolus - Northeast Ridge Route to the connecting ridge between Eolus and North Eolus - Photo #2. Scramble up to the ridge. If you're climbing Eolus first, cross the Catwalk, summit Eolus and return to the notch in the connecting ridge - Photo #3 and Photo #4.
Scramble up the North Eolus' south ridge on grippy rock - Photo #5 and Photo #6. This is a very short scramble. As you approach the summit, the pitch increases but the difficulty does not exceed Class 3. In fact, it's pretty easy Class 3. Once on the summit you'll have excellent views of Eolus, Sunlight and Windom - Photo #8 and Photo #9.
IMPORTANT: This route enters the Weminuche Wilderness area. Wilderness areas have special regulations and restrictions for party size, dispersed camping, campfires, etc. Also, dog owners should read the wilderness information carefully because some wilderness areas prohibit dogs to be off-leash and/or limit how close dogs can be to lakes and streams. If you have questions about the Wilderness area, please contact a U.S. Forest Service office for the National Forest(s) listed above.