I've been up Lyell, but I am no authority on it.
The normal route up Lyell involves leaving the JMT somewhere above Lyell Base Camp, then circling around to the right under the Lyell Glacier. If lucky, the bergschrund will be open so that you can see it and avoid it. If it is covered with a thin layer of snow or ice, it could be a death trap. Once you get under the rock wall, it is a steep rock scramble, although I remember a pitch that was awful with new snow on it. There may be other routes to the summit, but they will likely involve a much longer approach route.
The only problem we had was after we came off the summit and backed down to the top of the glacier. Two of us started to walk around clockwise to retrace the way we had come up. One guy started directly down the glacier, and he had no crampons for the blue ice. Worse, he was carrying an ice axe, but not in the Ready position. Needless to say, he was falling out of control after about 50 steps. He got beaten up pretty badly by the time he stopped halfway down the glacier, so he limped off the bottom under his own power. Kids, don't do this at home!