I'm seeing lots of Sierra advocates posting here, but nothing on the Rockies.
It probably won't take five weeks, but the Highline/Fremont (aka Continental Divide) trail through Wyoming's Wind Rivers from Big Sandy Opening to Green River Lakes is outstanding. There are many side trips you could take to expand the 80-mile point-to-point journey. One of these would be to visit the Cirque of the Towers and then take the Lizard Head-Bears Ears trail loop, returning over Hailey Pass. Farther north, you could take the Hay Pass trail to Golden Lakes and the Alpine Lakes. If you're skilled in cross-country travel and have the skills and equipment for glacier travel, you could exit this area via Indian Pass and Titcomb Basin (if not, exit via Hay Pass or the nearby Fall Creek Pass). A trip to Titcomb Basin is also mandatory. At the north end, you can visit Peak Lake and/or take the Doubletop Mountain trail west from Summit Lake, descending to Green River Lakes via the Porcupine Creek trail. You'll be above timberline most of the time, and the weather can sometimes be brutal (lots of frosty nights and snow, even blizzard conditions, possible at any time), so be prepared accordingly.
There is a shuttle (somewhat pricey) run by the Great Outdoors Shop in Pinedale, WY, which makes a one-way hike easily do-able.
You could spend 2-3 weeks here and the rest of your time hiking another stretch of the Continental Divide Trail. Or you could consider the Colorado Trail, which I believe would take close to a month.
If you have to start in late June, though, note that the meltout for these trails in a normal snow year (which this year seems to be) is early to mid-July. You'll be traveling on lots of snow and having to ford a lot of swollen streams. If it's possible to postpone the start of your trip until, say, the second week of July, you'll have better conditions.