Black Kaweah is a tall, remote, classic peak of the high Sierra. The volcanic composition of the Kaweahs starkly differs from most Sierra destinations. The easiest routes are class 3 and feature more loose rock and rubble than most climbs. It's well worth the trek and climb, though, as the summit views are phenomenal and the peak still has its original summit register from the 1920s, which is not even half full and has entries from all manner of Sierra notables including Norman Clyde, Walter Starr Jr. (signed in his own blood!), and the authors of all your favorite guidebooks!
One reason Black Kaweah is seldom climbed is its remote location. Many have dayhiked the peak from Mineral King, but all of the (one day) approaches should be considered strenuous. The peak can be approached from Mineral King (the closest point you can drive to) or from one of the trailheads in Sequoia proper (further but less rugged terrain to cross). From Mineral king, the two most common routes are to first cross Glacier Pass and then alternately traverse to Blackrock Pass or cross Cyclamen (or "Hands and Knees") Pass to the Little Five Lakes basin. From here a trail leads to the standard basecamp near the old cabin in the Big Arroyo.
The standard route, rated class 3. Many have rated this class 4, but (at least according to Patrick) with the correct route the exposure can be kept to class 3.
Rated class 3 by Secor, who claims this is the easiest route, though not as direct as the SW Face. This route follows the West ridge of the peak until a notch bars progress, then descends a couple of hundred feet to the top of the 'giant ramp' mentioned in the SW Face route description above. The final section follows the SW Face route.