Mt Donna Buang is the closest and cheapest way to see the snow from Melbourne but that does come with some downsides too - it can be really busy and crowded on big snow dump weekends so come prepared to wait in traffic or walk a little more than usual. Bring/hire snow chains if it's a decent dumping of snow, lots of people get stuck and have to be dug out so don't let that be you!
If you’re coming from the 10 Mile Turtable Car Park you can take a 1.2km walk through the forest to get to the summit or if the road is closed you’ll have to take a decent hike up Donna Buang Road. When you get to the summit at 1250 metres above ground, you’ll find a 21 metre lookout tower (closed in snow conditions), toilets, an undercover BBQ shelter and lots of room to make snowmen and toboggan (just make sure to BYO as there are no facilities here to hire)!

If you're visiting outside of the snow season you'll be able to drive right up to the summit without any issues and access the top of the fire tower for views of the Yarra Ranges.

I use these two public Facebook pages to work out snow conditions (locals usually are nice enough to share conditions):

https://www.facebook.com/groups/1711024152610126/

https://www.facebook.com/groups/mtdonnabuang/