Have you checked all fuses? When you turn the ignition switch to the "ON" position, is it holding a red light on the dash for the immobilizer?
If no red immobilizer light staying on, possible bad ignition switch or activation rod from ignition switch to key switch problem.
Or bad starter, starter relay, starter solenoid, dirty battery cables, etc.. all these items need to be checked!
There are some useful troubleshooting guides (pdf files) in the Factory Service Manual located at:http://colorado4wheel.com/manuals/Jeep/KJ/http://www.colorado4wheel.com/manuals/J ... Manual.pdf
STARTING SYSTEM page 8F - 33, DIAGNOSIS AND TESTING - STARTING SYSTEM
As to the P5013 DTC:
P0513 DODGE - Incorrect Immobilizer Key
Faulty immobilizer key
Poor electrical connection for immobilizer circuit components
Faulty immobilizer control unit
Faulty immobilizer key antenna
Read more: http://www.engine-codes.com/p0513_dodge.html
2005 Jeep Liberty CRD Limited
Blk BrushGuard, IronmanCoils/Bilstein's, Yeti Stage4 MaxTune, Weeks ElbowKit &Battery Tray,
No FCV/EGR, Samcos, ProVent, SunCoast&Transgo, Carter intank-pmp, 2mic. Sec. filter,
Hella's, Flowmaster/nocat, V6 Airbox, GM11 blade fan & Hayden HD clutch
98 Dodge Cummins, 5"exh. Viair, HugeK&N