East Buffalo Peak is sequenced with West Buffalo (Marmot Peak). It would make little sense to climb only one of these twin summits. There are also several ways these peaks can be approached. What appears to be a preferred route for Front Range peakbaggers is to drive in eight miles on FR471, which is 12.4 miles south out of Fairplay on US285. The trailhead used for that access is the Lynch Creek. There is also a direct access out of Buena Vista on a combination of County and Forest Service Roads 371 and 375 to the Four Mile TH. There are multiple reports of these other routes on SummitPost and 14ers.com. The access we used in 1995 offers a third option but is best done with 4WD. While we did not find the road access to be particularly rugged, a 4WD vehicle would offer the most assurance of making it to the old mine site near 11,320 ft. Also be aware that there are some private property inholdings in this area and respect them, especially if camping.
From the car park, probably at or near the old mine, continue following the old roadbed as it heads north, then west to gain the broad south ridge of East Buffalo Peak. Once on the ridge, follow it to the east summit on mostly grassy tundra observing the enormous giant thistles that seem to prefer this area. Higher up the tundra will give way to steeper & looser rock, kind of a combination of larger boulders somewhat embedded in the grassy tundra and then, even more loose rock on a steeper slope higher up. Both summits are volcanic in origin. You may want to identify volcanic features as you make this hike. There may be game/goat trails that will lead through the rock. As you near the summit ridge, the angle of ascent will become more tolerable and you can finish on tundra to the large, somewhat flat summit covered in tundra and embedded rocks. From this summit, most will want to continue northwest to West Buffalo Peak.
If you don't like the direct ascent up the south ridge of East Buffalo, it is reportedly easier to head toward the saddle between the two Buffalo summits. From the saddle, follow the connecting ridge to each summit, then return by the same approach. This will involve more mileage (not a significant amount) & contouring.
Links to other information, routes & trip reports for this peak that may be helpful.