Because this is not the fastest way to the top of the Matterhorn from Twin Lakes, this climb is normally done in conjunction with other hiking/climbing scenarios in the region. It definitely is the most enjoyable way to the top from Piute Canyon or the back/SW side of the Sawtooth Ridge.
This climb is mostly fun, solid 3rd Class climbing with plenty of opportunity for variations and increased difficulty on quality rock. When approaching from the NW, the face appears to be broken up into several chutes. The chute on the right of the one leading directly to the summit is likely the easiest (the large right-hand chute in this photo).
If taking this chute, traverse the low-angle slabs at the base, and maneuver directly into the bottom of the chute (or, take the first eighty feet or so of the main chute leading to the summit and then traverse between some towers on the right into the right-hand chute). Climb directly up the chute, and take the path of least resistance to the summit ridge (I found this to be gradual climbing to the left, but there are many possibilities).
Hang a left at the summit ridge and hit the top!
Reverse the route, or descend via the East Couloir to Twin Lakes.
No gear is really required for summertime climbs. If any in the party is apprehensive at all about 3rd Class climbing, bring a rope and pro along for good measure, as getting off-route could make things a little exciting.