Just by way of introduction, you need to remember that the weather that hits Tasmania's southwest has travelled unhindered across vast stretches of ocean. It arrives in a hurry, and can be very cold and wet. The wind can often be strong enough to knock you over. On the mountains accessible from the Huon, the weather isn't normally quite as bad as on some of the ranges closer to the Southern Ocean (e.g. Precipitous Bluff, Ironbounds) but it can be very ordinary.
For example, Hartz Peak is a quite short walk, but you can find yourself in freezing sleet at Hartz Pass even though it was sunny and blue when you started out from the car park.
The Bureau of Meteorology (or here for Hobart and other cities) is a good place to start. There's an automatic weather station at the Hartz Mountains, and the readings are available here. This is a reasonable guide to weather at altitude across the Huon area. If visiting, you can see the weather station at Keoghs Pimple.
Tasmanian weather's sometimes almost as bad as Melbourne weather!
The most important thing to know about Tasmanian weather is that it can be very unpredictable. People think it's cold and wet, but that's not always the case. However, at times it is, and like in Melbourne, sometimes it changes very quickly. This can tend to catch out people from elsewhere, who assume there to be no danger on a sunny day. If the weather changes while you're at the far end of your walk, it can make for a very uncomfortable or even dangerous return. Furthermore, you need to account for the potential to be delayed, by slow party members, illness, injury, getting lost or having to assist someone else. Especially at high altitude, you should assume the need to be warm and dry, protected from cold wind and rain. Hypothermia can occur even on relatively warm days if wind and rain are present.
Having said that, Tasmanian weather is often superb for walking. Summer is normally cool to warm, sometimes wet, sometimes hot. Overall summer in Tasmania is far more comfortable from a weather point of view than many other places in Australia. Sometimes summer starts in December, but often the real warm weather is January to March. Autumn is believed by many Tasmanian walkers to be the best walking season. Days are often still although cool, with crisp clear mornings providing great visibility. Winter is obviously riskier, and is normally cold to cool, and often wet. It snows often at higher altitudes, and occasionally all the way down to sea level. You need to pick your day and suit the walk to the weather. Take some local advice, and remember the days are short. Spring is often wet, and in fact the days can be just as cold as winter. Snowfalls can continue quite late (and in fact at any time), and a recent fatality in the snow occurred at Christmas. The days lengthen, but many tracks will be wet underfoot.
Tasmanian weather is predominantly westerly. Westerly weather normally works like this: A cold front approaches from the west. The wind blows from the north or northwest, bringing rain. The wind then gradually swings around towards the west, then southwest and perhaps south. At some point, the rain stops and as the wind swings it tends to get colder. Then the wind eases ready for the next front. Sometimes the pattern happens very quickly, and sometimes it's interrupted. Easterly weather happens less often and is less predictable. It can bring very heavy rain.