If the sun wakes you up you are already too late in the morning. Expect a wait. If you stay at the festival through a headliner you are already too late. Expect a wait. The wait times get progressively longer each day as people tend to hold out the first day or two.
The best place to shower and the best time would be as follows: Go to the south by south-easternly most shower. Wait there until the chicken goes "koo koo ka-chaw" shortly after a shooting star crosses in front of Venus just before the dawn. Then stand on one foot and wiggle your left pinky on your right hand. The Coachella veteran that sees you do this (the famous shower signal) first will invite you to shower with them and take you to the front of the line. You're welcome.
Do it during a headliner or set an alarm for a few minutes before they open them up. It may suck to be up that early, but it'll be worth it to be able to be clean and not pissed off and dirty waiting in a line.