The usual approach is from Leavitt Meadows. Follow the directions on the main page to Tower Lake.
From Tower Lake, head to the obvious saddle to the south (Tower Pass); a good use trail can be found leading up to the pass. Two permanent snow fields are found here; in middle or late season, these can be bypassed easily, but in early season, it may be easiest to contour east over class 2-3 rocks to their side.
From Tower Pass, head southeast through grass, slabs, and some small granite benches up the ridge, bypassing another large permanent snowfield (shown on the 7.5'). The left side of the snowfield is fairly stable talus and marginally more direct, while the right side is easy slabs; neither choice seems clearly better. Either way takes you to Tower's north ridge.
Judging from the sandy use trails seen below, after bypassing the large snowfield, many people seem to drop down from the ridge and traverse across Tower's NNW Face (class 2) before making their way up the NW chute. However, the climbing is more enjoyable by keeping to the spine of the north ridge (class 3 in places, on solid rock with good holds).
Eventually, following the north ridge leads to towers, and it is necessary to head west around these to angle left up a chute to the summit. This is the "staircase chute" referred to by Secor, and it may be confusing at first; it appears to be unambiguously class 2, not the class 3 widely reported in guidebooks, and that's basically what it is--there is a move or two of what might be considered class 3 at the top, but nothing more.
suggests the following shortcut:
As the snow melts back from the Sierra crest in mid to late summer, one can shorten the climb to Tower Peak by bypassing Tower Lake and Tower Pass. While in the meadow below Tower Lake, look south to the snow chute on the right (west) side of the Tower. There are two branches. The left branch will have snow longer and is a formidable climb. The right branch is mostly out of view, but if it looks at all like shown in this picture
, you can find an easy class 2 route snow-free to the Sierra crest.
None required. An axe and crampons might be needed in early season to negotiate the chute.