Both routes start from the North Lake trailhead west of Bishop.
(Beware that the topomaps of this region are still in the ancient imperial system, which means that they have not been revised in decades)
Southern RouteHike to the Lamarck Pass (no need to climb Mt Lamarck itself) and then head down the southern side into Darwin Canyon. Follow the many lakes downstream until you reach a point where you can head north (turn right). That's Darwin Bench. Hike up towards the lakes of this valley and you'll find two lakes at about the same elevation. Walk in between them and start ascending the class-2 slope to the summit.
Northern RouteTake the Paiute Pass trail from the North Lake trailhead and make sure you don't take the left fork that goes to the Lamarck Lakes. Continue to Piute Pass. A use trail continues to Muriel lake. Coast the lake going west and then south, and you should reach Goethe Lake. The western shore of this lake is easier. Coast it south and then, past the lake, head southeast to Alpine Col, the pass between Mt Goethe (right) and Mt Muriel (left). Follow the ridge east from Alpine Col to the summit via a number of false summits that involve many class-3 moves, or, better, cross over to the south side of Alpine Col and then turn right (west) because this side of the mountain is mostly class-2 walking.