The Whitney Portal Campground is a great place to camp and do some hiking...... also consider, Horseshoe Meadows, Onion Valley.
Each has its own beauty. Reservations can be made 6 months prior to the date you want. Whitney Portal and Onion Valley Campground have both reserve and first come sites.
You can reserve or check out info below.https://www.recreation.gov/
Horseshoe Meadows is walk-in....and at 10,000 feet. Whitney Portal is 8,000 and Onion Valley is at 9,200 feet.
There are other campgrounds, like Lone Pine or Grays Meadow, however they are at lower elevations and get rather hot during the summer. I've been at Lone Pine when it was close to a 100. Not fun.