Here are a few ideas, all on the west side of Ross Lake:
No. 1 -- Loop trip (pretty much, but some backtracking to return from Whatcom Pass): catch a boat ride from Ross Lake dam up the lake and get dropped off at trailhead at Little Beaver Creek outlet on west side of the lake; hike westerly along Little Beaver Creek trail to Whatcom Pass; then return back down from Whatcom Pass to exit via Big Beaver Creek to Ross Lake -- making sure you remembered to arrange for that boat from Ross Lake dam to return and pick you up on the day you expect to arrive at Big Creek's outlet on Ross Lake.
No. 2 -- Another "pretty much" loop trip: enter from trailhead at Ruth Creek on west side of North Cascades Park; hike up over Hannegan Pass, then gain the crest of Copper Ridge and follow the Copper Ridge Trail traveling north; before reaching Canada, drop down from the ridge to the Chilliwack River valley; hike southwesterly along that river valley to junction with Brush Creek; hike southeasterly up Brush Creek to Whatcom Pass; then retrace back down from that pass to the junction of Brush Creek and Chilliwack river; continue southwesterly on the trail along Chilliwack River back to (and over) Hannegan Pass; and exit back to the Ruth Creek trailhead.
No. 3 -- Traverse (no loop; requires dropping a vehicle at one end or other arrangements): combine above two routes by making arrangements for a car to be waiting at Ruth Creek trailhead on west side of the Park, and then get to Ross Dam and take that boat from the Dam up the lake to the outlet of either Little Beaver Creek or Big Beaver Creek (the trail up either one of those creeks takes you to Whatcom Pass); hike to Whatcom Pass; cross the pass and hike down Brush Creek to Chilliwack River; then either:
Option 3A --- at Brush Creek junction with Chilliwack River, go northeast on Chilliwack River trail to Copper Ridge trail; and climb switchbacks up to Copper Ridge; hike southwest along crest of Copper Ridge to Hannegan Pass to Hannegan Pass; then over that pass and down the Ruth Creek trail to exit at its trail head; OR
Option 3B -- at Brush Creek junction with Chilliwack River, go southwest on Chilliwack River trail to Hannegan Pass; then over that pass and down the Ruth Creek trail to exit at its trail head;
where, waiting at the Ruth Creek trail head, you will find (hopefully) the transporation you arranged in advance.
There are excellent books for North Cascades NP trails, and "WOW Guides for North Cascades" by Kathy and Craig Copeland looks like one of them. My trips in the above areas predate (by a couple of decades or so) the above guide book, which had its first edition in May 2007, but it has clear maps (sketch-type diagrams) and great photos -- although it doesn't seem to have much info on Little Beaver/Big Beaver Creeks.
A huge draw for all the above trips is the spectacular view from Whatcom Pass of Mt. Challenger and its hanging glaciers at the head of Beaver creeks, and expanding views as you hike up even higher above that pass to Tapto Lakes.
There are also spectacular views of entire range while on the Copper Ridge Trail -- also with more terrific views if you scramble from Hannegan Pass to the top of Mt. Hannegan.
I haven't done the traverse described above, but did the two end loops. However, the link pasted below from nwhikers website provides a great trip report of the traverse (solo). I don't know the author, but he sure did an outstanding job with the traverse, and with the story -- very nice photos to boot: