Generally in the inner canyon you can expect temperatures around 70° F or warmer. During the day, you can expect temperatures around 100° F or warmer. June is typically one of the hottest months in the Grand Canyon; July and August are more humid and have afternoon thunderstorms that block the sun but are still nice and toasty warm.
Go here http://www.nps.gov/grca/planyourvisit/weather-dangers.htm for some statistics.
Plan on having reserve water and plan on drinking a lot of water to stay hydrated. It is hard to tell when you are drying out in the dry air there. There is plenty of water on the Bright Angel Trail between the south rim and the river. There are several water sources between the river and the north rim but you will be climbing another 1000'. When in doubt, carry more water than you think you will need and keep to an easy pace. It is the young hotshots that generally get in trouble in the canyon.