Australia has a surprisingly good sized winter sports industry. The season is not very long compared to more 'favourable' climes, but Australians take full advantage of what they've got, extending the natural ski season with snow machines, etc.
The largest and highest alpine skiing resorts, usually having the most reliable snowfalls are found in the mountain areas of New South Wales. Victoria also has extensive ski fields along the ranges. Tasmania is located much further south and has more extensive snows during the winter, which make for some winter hiking and cross-country skiing opportunities, but alpine skiing is more limited, with a shorter season and ski smaller ski resorts that are harder to access.
In News South Wales and Victoria the ski season traditionally begins on the Queen's birthday long weekend (the second Monday in June is a public holiday). The beginning of the ski season is more often marked by a celebratory drink than a early morning run down fresh powder. The most reliable snowfalls are in July and August. If you had to pick one week of the year, the second week in August would have to be the choice with the best chance of good alpine skiing. In good seasons there will be some lifts open in June, September and sometimes early October.
(Thanks to WikiVoyage.org)