You aren't going to be able to walk from LA to NYC in a week even if you don't stop to sleep. If you walk 24 hours a day and never sleep (which is impossible since a human can't go a week without sleep) would require you to go almost 15 miles per hour. To make the trip in one week you would have to constantly run 4 minute miles (even olympic level runners can't keep this up for more than about two miles) all the way there and never stop to eat, sleep, or drink.
I'm not saying you can't walk from LA to NYC; people do that all the time. I'm just saying a week isn't enough time to do it.
Also, are you planning on bringing a tent and just going backpacking or are you planning on renting hotel rooms? I doubt it is legal for you to rent hotel rooms if you are under 18 without a parent's consent. Also, $600 won't be enough for hotel rooms.