Follow the Eagle Peak trail upwards from Longmire (~2800') on a well-graded and well-maintained series of switchbacks for ~3.6mi to the saddle between Eagle Peak and Chutla Peak. Elevation gain to this saddle is ~2800'.
From the saddle, head south along an obvious user trail which seems to dead-end shortly atop a steep cliff. Back up a few feet and look to the right to locate a user trail concealed by the trees/bushes that drops down throug the trees to descend below the cliff face. Continue along the ridge following the user-trail up and down - bypassing the occasional high point along the ridge and making your way towards Chutla's summit. The route does not require any gear, but extensive use of the "veggie-belay" (e.g. roots & branches) is probably recommended. There is a bit of slightly exposed rock scrambling along the way (class 3-4).
For primo summit photos on a good day -- have folks stay on the summit and walk down the southern ridge a short distance for terrific background views of Mt. Rainier.