Half of Backpacking Mt. Whitney is having hiking experience. The peaks you mentioned are great preparation peaks. You work your way up to being ready to hike Mt. Whitney.
The 2nd half is the elevation thing.That can be a real challenge the first time you are above 12000 feet. Spend as much time as you can hiking above 10,000 feet before you go to Mt. Whitney. Even a few hikes within a month of Mt Whitney will really help.
Then spend 2 nights at the portal, followed by two nights on the trail. One night could be at Outpost camp and one night at Trail Camp. On the third hiking day, summit Mt. Whitney from trail camp. This will minimize the elevation problems.
The best time to join a group might be in the summer, when people fall out and can't come. During June, July and August many groups have an open slot. Now might not be the best time to run into an available slot, since no one knows if they even have a permit yet. If you didn't become part of a group, you could always try to pick up a no show permit. With all the people on the Mt. Whitney trail, you would never need to feel alone.
If you follow this plan, you should be fine and have little to worry about. Good luck.