June to September is peak season at Berchtesgaden: the Königssee boats run at full frequency, the Kehlsteinhaus is open (mid-May to late October), and the hiking trails are clear of snow. The downside is crowds, particularly at the lake and the Eagle's Nest. July and August can feel overrun on weekends — midweek visits are significantly more pleasant.
The shoulder months of May and October are arguably the best times: fewer visitors, the Jennerbahn and boat services running, and the mountain light spectacular on clear days. The Dokumentation Obersalzberg museum is open year-round except Mondays (and December–January).
Winter brings snow to the Jenner ski area (December–March), and the landscape around the Königssee, while the boat service reduces to its winter schedule, is exceptionally beautiful under snow.