you cant just hook up a deep cycle battery to a car alternator. the charging method is completely wrong for a deep cycle.
car batteries are designed to be constantly charged by the alternator. deep cycle batteries are deisgned to be run down without recharging, and then fully recharged with a regulated supply.
if you are planing on throwing a deep cycle in the trunk with a recharging line from the alternator, then you are going to looking at a problem.
you will have to rig up something that will regulate the charging scheme for the deep cycle, and i have no clue what you are going to require to recharge that battery.
simply put. it wont work exactly as you have described it, but it could work if you rig it up correctly.