MTB strongly recommended; XC with racks a great idea. Time of year is a conundrum -- there is a chance of bush fires between Christmas and the end of February when the really hot and dry weather with strong hot northerly winds occur. Before then can be cool and windy with some rain. After then... well it's probably the best time because the schoolies have gone, and the weather is mild to warm, but can be cold in the higher altitudes.
Take real care in selecting the clothing you bring along; especially pack enough to keep you warm in snowy and wet conditions. Wool and layering with a quality waterproof jacket are good. Same senitments apply to tents. Tasmanian weather has a well-founded reputation for being unpredictable -- snowfalls on Mt Wellington overlooking Hobart in late January are not unknown.
I haven't done the trail, but was involved on the periphery of its administration, and know many of the areas that it passes through as a result of cycling and car rallying. My former randonneuring riding buddy has done the trail, and he reports lots of climbing (estimated around 40,000 feet total), and much of steepest and roughest stuff is in the north-west. The scenery throughout is breathtaking and highly varied.