Start your hike at Lake Eibsee. The lake is a great place to be at dawn, and it's fairly uncrowded early in the morning. There is a campground nearby and a parking lot at the lake, as well as the cable car station, so you'll end at the same point if you decide not to hike back down.
Head south on the marked trail toward the mountain, located just behind the paddle boat dock.
Pass an Austrian border crossing sign along the hike. The path should shortly loop back around into Germany. There are no border guards here, as both countries are EU members.
Head through the Hollentalklamm gorge. You will know you are there when you reach the cable car lines again. Along the cables you will pass the "Hollentalangerhutte," where you can get a meal, beer or bed for the night.
Cross the Hollentalferner, the remains of a small glacier.
Climb the rock wall up to the Irmerscharte, the last gap before the summit. Use the iron handles and ladders on the wall. Climbing gear is recommended but not necessary.