The next step would be go go back into AirPort Utility and temporarily set the wireless security to None as a test.
Then, see if the DirecTV product will connect that way.
If it will, then you know that you have a wireless security compatibility issue with the DirecTV device and AirPort.
My DirecTV device does connect fine using WPA2 Personal Security, which is an industry standard.
However, if the DirecTV product will not connect with "no security".....it's a no brainer to know that it will never connect "with security". Got to suspect a defective DirecTV product in that case, but DirecTV will still likely blame things on the wireless router.
I've had iPhone issues in the past that were solved only be restoring the iPhone back to its original settings. Work you way through the troubleshooting guide for iOS first, though.
http://support.apple.com/kb/TS1398
The real iPhone experts are over in the iPhone support area:
Using iPhone
Please keep us posted on your progress.