4-5 days isn't very much time so I agree with focusing on one or two areas or do a north-south road trip (or visa versa) either picking the east coast or the west.
There is tons of hiking, although a lot is remote and/or multi-day and bush walking can be quite hard (think 'routes' through waist deep bogs and climbing over boulders). Cradle Mountain is a lovely area with accessible day walks but you won't see much if the cloud is in. Similarly on Maria Island, where the walks to Mt Maria or Bishop & Clerk are best done on clear days. The Tasman Peninsula area has plenty of day walking, but there is good bush walking all over the state, even in the 'populated' areas around Hobart. Tastrails.com website gives a good overview.
If you want to see wildlife, you are best to stay at locations outside of major towns and/or where the owners don't fence off their gardens! There are lots of rentals (affectionately known as shacks) in bush blocks to rent, where you are likely to see pademelons/wallabies and smaller macropods and brushtailed possums at night.
If by 'unique fauna' you mean devils, there are specific accommodations that feed them so they are your best bet (there are two places I can think of in the north that can virtually guarantee wild devils - there are many wildlife sanctuaries who also offer tours). They occur all over the state and they travel large distances within territories, so seeing them is just luck. The devils on Maria Island are a released population that had become accustomed to humans, part of the devil tumour breeding program. The population has settled in somewhat and are less often in the campsites these days, but they are still sometimes seen. You will see wombats on Maria Island if you stay overnight.
One itinerary good for mountains, wilderness and wildlife would be Hobart-Mt Field-A10-Strahan-Franklin Gordon-A10-Cradle Mountain-Launceston. That could take 5 days to get you back to an airport. With more time, you could head to La Trobe one early morning or evening to look for platypus (they are also present in Mt Field). If you haven't had your fill of macropods, narawtapu has large mobs of kangaroos around Springlawn every evening, as well as wallabies, possums, eagles and snakes. The main thing is to get out early or around dusk. But please do not aim to travel at night, or be prepared to drive really slowly. The wildlife really is everywhere and it's not very fast at crossing roads.
Alternatively do a day trip from Hobart to Tahune and a walk in Wellington Park to see the mountains of the south-west in the distance, then head to the Tasman Peninsula (hiking, cliffs, wildlife, heritage at Port Arthur), spend at least one night on Maria Island (more walking and wildlife) and enjoy some good wine and food along the east coast. Unfortunately March is the wrong time of year for the penguins at Bicheno but otherwise I can recommend the tour (they come right up through the walkways in breeding season when they are feeding their chicks).