Numerous approach options exist. One is to hike up the Budd Lk. trail and then skirt the Echo Peaks on the right (west) before dropping down into the valley west of the Matthes Crest. Then ascend to the slabs north of the North Ridge. Alternatively, go through one of the cols in the middle of the Echo Peaks. Another option is to first climb either the Cockscomb or Echo Ridge. The approach from either of these formations is straightforward.
Approach by hiking from the easy slabs just north of the Crest. One can easily bypass the initial two towers by hiking and scrambling on slabs to the west of the Crest (these two towers are uncharacteristic of the rest of the climbing and are frequently skipped). The climbing is initially class 3 and soon becomes class 4 as one attains the ridge. The routefinding is too intricate to detail here and in any case would spoil your fun. Suffice it to say that you need to be imaginative about where to go. The route wanders from side to side and on one occasion drops down significantly on the west side. Staying on the very top of the Crest the entire time would make the climb much more difficult. There is a particularly enjoyable section where one hand traverses on a slab that curls around underneath like a breaking wave. Delightful and pretty easy, but difficult to protect.
As one approaches the summit, the climbing changes from horizontal to a gentle incline. There is some interesting climbing here to match the routefinding. In one spot, a gap must be negotiated. There is significant exposure as one approaches the summit.
Note that although the climbing is never particularly difficult (provided one is careful with the routefinding), the lack of adequate protection, the nature of traverses, and the sometimes crusty nature of the high country granite make this route a poor choice for towing along an inexperienced climber. It is best that all climbers be comfortable on class 4 and easy class 5 without adequate protection.
One can rappel from the summit with two 50m ropes or probably with one 60m. A much more enjoyable choice is to retrace your steps.
Light alpine rack, nothing bigger than 3". Numerous slings to reduce rope drag. Long slings (doubles) or cordalettes to loop projections are useful. There are many sections that are not easily protected.
There's very little detailed published information about this route and one can understand why: it's a difficult route to describe. Most guidebooks focus on the South to North traverse, but have little to say about the second half of this climb, which is a shame since many people consider the northern half to be better.
The Good, the Great, and the Awesome (Peter Croft) -- A nice description of the S-N traverse.
Climbing California's High Sierra (Moynier/Fiddler) -- A very brief description of the S-N traverse.
The High Sierra: Peaks, Passes, and Trails (R.J. Secor) -- Completely ignores the north half of the ridge.
On the web: Supertopo -- Supertopo sells a PDF (as well as a print version) of a Tuolumne select guide that includes the Matthes S-N traverse including a few details about the northern half.