AN MD-80 CREW MISREAD CLRNC DURING AFTER LNDG TAXI; STARTED ACROSS RWY NOT CLRED TO CROSS AT 'KCHS.'

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AFTER LNDG AT KCHS ON RWY 33; HAD TO TURN OFF AT END OF RWY; DUE TO TXWY 'E' BEING CLOSED FOR CONSTRUCTION. LOCAL CTLR TOLD ME TO TAXI VIA AFB MILITARY RAMP AND TXWY 'H'; AND TO HOLD SHORT OF RWY 3 (I THOUGHT HE HAD SAID HOLDSHORT OF RWY 33). MILITARY SIDE OF ARPT HAS NO SIGNAGE WHATSOEVER. ALL TXWYS; RWY HOLDSHORT SIGNS DO NOT EXIST. ON TXWY 'H'; I STARTED TO CROSS RWY 3 WITHOUT CLRNC. AS I WAS XING RWY; CTLR STATED 'CLRED TO CROSS; CONTACT GND ON OTHER SIDE.' UPON CONTACTING GND CTL; CTLR SAID NOT TO WORRY ABOUT THE XING-THERE WOULD NOT BE A PROBLEM. OVERALL; I REALIZE IT WAS MY FAULT FOR XING WITHOUT A CLRNC. HOWEVER; I THINK I THERE WAS SOME FORM OF HOLDSHORT SIGN ON TXWY 'H'; IT MIGHT HAVE PREVENTED RWY INCURSION. THE FAA NEEDS TO HAVE AIRFORCE INSTALL TXWY AND RWY SIGNS ON THEIR SIDE OF ARPT; IF CIVILIAN ACFT ARE GOING TO BE TAXIING OVER THERE. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: AKCHS TWR SPECIALISTS STATED THAT THE CHARLESTON AFB OWNS THE ARPT PROPERTY AND CONTROLS MOST OF IT'S AREA; INCLUDING THE W AIR FORCE RAMP AND TAXIWAYS. 'DELTA' AND THE AFB RAMP ARE NORMALLY NON MOVEMENT AREAS. HOWEVER; WHILE TXWY 'E' WAS CLOSED DUE TO CONSTRUCTION; PERMISSION WAS GIVEN TO USE 'D' AND THE AFB RAMP AS A TAXI RTE TO AND FROM THE TERMINAL. THIS AREA (THE W SIDE) WAS RPTED BY THE SPECIALIST TO BE LACKING IN PROPER SINAGE. TAXIWAYS A; B; AND G (THE E SIDE) ARE CONTROLLED BY THE 'CHARLESTON COUNTY AVIATION AUTHORITY' AND DISPLAY PROPER SIGNS AND MARKINGS.