How would you go about a planning a backpacking trip into the Grand Canyon National Park.

Suppose you wish to camp anywhere in the park other than in developed campgrounds on the North or South Rim. In that case, you must obtain a permit from the Backcountry Information Center. Camping sites are limited in the Grand Canyon, and spots are issued four months before a given month. E.g., For January, permit applications should be submitted between Aug 20 and Sep 1 to be considered "earliest requested."
For requests received during the Earliest Consideration period, the day received does not matter; all Earliest Consideration requests received during this span of dates are treated the same. It may take Rangers up to three weeks to process all Earliest Consideration requests.
Subsequent Requests: Once all Earliest Consideration requests are processed, Rangers will begin processing written requests received on subsequent dates. These Subsequent Requests are kept and considered in the order received. Submissions will be accepted as late as two weeks before a hike start date.
 
If your permit request is selected, you must diligently plan your trip. You will need to carry everything you need for the duration of your journey with you except for water which is few and far between. Know where the water sources are on your route and carry filtration devices, water purification equipment, and backup equipment. The hotter the trip, the more water you will require. Have a plan for where the necessary water will be on your route, and double-check with the park service ahead of time that water is still there. Dehydration is the number one reason for rescue and death in the canyon. Some springs dry up in the summer months.
 
Travel as light as possible; you must pack everything you bring into the canyon, including used toilet paper.    Plan each meal, knowing how much calories and water you consume based on your planned exertion level. Calorie and water requirements also change based on temperature (the hotter the day, the more water and food you will need). Also, make sure you have the appropriate gear based on temperatures. It is approximately 20 degrees cooler at the top of the canyon than at the bottom because there is a 5000 feet vertical difference at the South Rim and 6000 at the North Rim. Ensure your footwear is "broken in", and carry first-aid kits with particular attention to foot care, such as blister pads and moleskin.   
 
Start with shorter, more leisurely trips; if you like them, move up to more adventurous multi-day trips. Plan on rest days when you can enjoy the majestic scenery of one of the world's natural wonders. Above all, enjoy the canyon; it's a magical place 6 million years in the making that exposes geology over 4 billion years old.