I don't think the direct Hume route is as boring as people think, especially on a bike. I quite like farm land. However, it wouldn't be my choice, just because of the noise and buffeting from the traffic (it's mostly freeway and the main truck route Melb-Sydney so most traffic is 110kmh).
Certainly, the coast route (Princes Hwy) would be more picture postcard pretty, with some interesting places along the way. That's essentially the link that MalP posted. To cut out south Gippsland and some distance though, from Melbourne, you'd head for the Latrobe Valley (Morwell, Traralgon, interesting power stations) to Gippsland (Sale). If you had more time, you could take a detour to Seaspray there, to have a look at the 90-mile beach.
I'd then take the highway to Stratford, and you can then ride two routes to Bairnsdale; the highway which would be OK, and if you had any mechanical problems you'd have passing traffic, or the back road via Bengworden, as per the link (but it's quieter). Seriously beautiful and steep downhill into Lakes Entrance (where you could usefully stay a night and walk over to the 90-mile beach). Nice town, with interesting fishing fleet and boat trips on the lakes as well. This (and probably Bairnsdale as well) would be a natural overnight stop.
Then it's nice country and bush, heading east to Cann River, and you then have two choices:
1, continue up the coast road, which is pretty, but not as wide as you'd like in places. Lots of coastal towns along the way. This would take longer than the direct route up the Hume, so 10 days won't be enough, but you could try to cut out some time with a bus leg, I guess. I think there's still a bus up the highway daily, but you'd have to book it in advance and talk to them about the bike. Don't know how that would work.
2, head inland up the quiet Cann Valley highway to Bombala, then Cooma, then Canberra (national capital). I haven't worked out the distances, but I think you'd need a bus/train from Canberra to Sydney as you'd be a bit short of time. This route takes you up into the high country, with real country people and country pubs, which is another side of Australia altogether. It's a fair way between towns (so plan your distances and mechanicals), but there's little traffic and it has its own remote beauty, IMO. One of my favourite places.
February can be very
hot - 30-40C maximum, although cooler at altitude, so you might want to factor that in to your mileage. Perhaps have a look at our Bureau of Meteorology site, www.bom.gov.au
to give you an idea.
Another option might be to cut out a day early in the trip, and just get a train to Morwell or Sale.
Good luck with your planning.