Determining the best time to start your ascent is a compromise. You want to start early enough that you minimize the chance of afternoon storms interfering with your trip, but you don't wan tto leave so early that you are fatigued and miserable at the start. (On some routes you might also want to consider the condition of the snow which can be easier to climb in the morning than the afternoon, but at this time of year it is not a factor on the main trail.) Assuming that you hike at an average pace, I would reccommend a start time of 4:00 AM plus or minus an hour. If you tend to do well with early morning wakeups, the afternoon weather may be stormy, or you hike slower than average then I would try to leave earlier. If you hike fast, the weather is expected to not be an issue, and you can sleep well the night before then leave later.
The lower part of the trail is the section you are most likely to be hiking in the dark, either in the early morning or later in the evening, is very easy to follow. While there are a few minor trails that spur from the main trail, they see so much less travel that most people may not even notice them. It is relatively easy to follow the trail with a low power headlamp or even by starlight. If you've not much experience with a headlamp I suggest a small training hike or walking around your house at night to make sure that the light beam is bright enough for you.
I'm sure you'll have a great trip.