The GCNP back-country is on a 4 -months- out written permit request delay
If you want to hike in July then you have to apply in March
Here's your site
http://www.nps.gov/grca/planyourvisit/backcountry-permit.htm
Here's the other half of it
http://www.nps.gov/grca/planyourvisit/overnight-hiking.htm
For what it's worth I personally don't think that anybody should be allowed to hike the Inner Canyon from 5-1 to 10-1
Just me
Mandatory reading
http://www.amazon.com/Over-Edge-Death-Grand-Canyon/dp/0970097301