Admir, first you need to get permits to hike it. With the Federal shutdown, Recreation.gov now shows "
Registration is Closed", and the lottery registration usually begins Feb 1. There is no guessing what will happen this year. But bottom-line is: You will first need to obtain a permit.
The camping spots you mentioned are pretty much the only places. Outpost is about 10.5K elevation, which is a safer zone than pushing up to Trail Camp at 12.5. Read up on AMS -- altitude sickness. Since you are coming from about 1k elevation, you need some acclimatization time. Spending a night at Whitney Portal, and then hiking only to Outpost and spending a night would be wise. Or you could spend one night at Horseshoe Meadows--you can drive there and camp, at 10k. Then hike up to Outpost from Whitney Portal. That's what I would do.
Then get up extra early and push to the summit from Outpost Camp. With two nights at elevation, you would be generally better prepared to get to the Whitney Summit.
On the other hand, if you really want to stay at 12.5k Trail Camp, you should spend two nights prior to that at elevation. Coming from so far, you need to do everything you can to give yourself enough acclimatization time.
As for hiking preparations: You need to spend some long days hiking -- just miles on your feet is good preparation. When I was younger, all the prep I did was jogging 12 miles per week, maybe a little bicycling added in. Some of that jogging included running around a stadium to get in some hill work. You just need to get in a good amount of aerobic training, and miles of hiking.
Ask more questions, maybe people here will add more to what I've written.
