Went North to South on the Kaibab Trails last Spring. Rode the hiker's shuttle back around.

Get a digital kitchen scale and weigh every item. Get a pop can alcohol stove if you must have hot drinks and oatmeal (can't haul any stove fuel on any airplane). In June, you won't need a real sleeping bag, but take some clothing for warmth during the night. You will have to layer up, later in the night. Consider bringing a light base layer to wear under your daytime cotton outer layers at night with a ground cloth that you can taco in. WalMart had some microfleece and some DriMore athletic wear on sale recently for $5 to $7. Regular fleece is too bulky to pack and flunks the weight test here.
Take jerky for meat. Leave the chocolate out of the trail mix. Take a lot of dried fruit for the potassium (bananas and apricots). Raisins are good for quick energy. Take some powdered Gatorade but don't overdose on it as it is much too salty when drinking a gallon a day. A liquid ounce per quart of water may be all you need for a sweaty day's electrolytes. Several small, cheap, store brand water bottles weigh much less than a few liter Nalgene bottles and the smaller ones don't slosh as loud when hiking. Take only as much soap, sunscreen, toothpaste, and bug repellant as you will use on your hike. Clean and reuse small empty containers or buy the tiny travel sizes. Bring white cotton gloves with the finger tips removed, for keeping the backs of your hands from getting so sunburned. Light cotton, loose fitting pants like hospital scrubs would be good trail wear. Choose a pattern that won't show how dirty your clothing gets from sitting on rocks for three days. The goal is to keep the sun off of your skin. From Phantom Ranch north, watch for shade in side canyons and under bridges. Wet your clothing and your bandana at every water source.

You are wise to use the time difference for starting early. At the top, a ranger praised my 8:00am exit until I mentioned my 3:00am start. Read up on condors. They perch just over the wall north of the South Rim ice cream shop behind the El Tovar Lodge. The standard joke is they are watching for ill-prepared hikers. The condors wear numbers so you can read their individual genealogy at the peregrine recovery website.

hitthetrail.com has some excellent info, stories, and Inner Canyon (Phantom Ranch) weather reports.