I'd start at Whitney Portal, forget the hostel, and would go directly to Trail Camp. This should be no more than a 3 day trip. If you have already backpacked to San G, you most likely already know how your body is going to react to going to 9,200' from sea level. Whitney Portal is under 8,400'.
However, if I had 5 days like you do, I'd go point to point starting at either Horseshoe Meadow or Onion Valley. I've done the former in 3 days; the latter in 4. They are both prettier trips.
