The deal is even with a .7 voltage drop from the alternator, the house battery should be keeping a voltage of about 12.8 for a substantial time and then slowly drops depending on the load. More load added obviously makes it drop faster. You're seeing 13.6 with the engine running so the battery should be getting a charge.
It's probably time to take voltage off the battery lugs themselves. As mentioned you should put a good hard charge on it for a day or so then test the battery voltage after no load or charge has been put on it for a few hours.
It sounds like a bad battery but a bad connection also can produce low voltage issues so you need to look over everything. Some batteries can be equalized and be brought back to better health. Each battery size type and manufacture will have different stats. You need to find the manufactures data on the battery. Here is Concorde's data:
http://www.lifelinebatteries.com/manual.pdf