(Sorry if this is a bit long. I felt like making more of a guide-like rant, because I hadn't really solidified my own tactics as Heavy. This helped.)
So you like Sasha too?
The only time the Heavy is pure win on offense is when he is ubered. However, that doesn't mean he is restricted to defense only when he's not -- even though he's listed as a defensive class along with the Demoman and Engineer.
I had some trouble conforming to a defensive role when playing the heavy, as we've been taught that generally the guy with the big gun is usually the first guy into the fray. So I insisted on playing him as a pure offensive class. As a consequence I was faced with mixed results; Sometimes I'd rip the opposition to pieces and win the day, other times I'd get mauled a new one. When the "pwn/suck"-ratio got too much in favor of the latter, I started suspecting that I was playing him wrong. So I looked at the other two classes he is grouped with:
The Demoman can be a nightmare to deal with if he has barricaded a chokepoint, or when he's retreating (chasing a demo is asking for trouble). When he's pushing forward he is at his weakest, but if he's smart about it he will push a little, retreat (when chased), then push again.
The Engineer with his sentry gun can be a real pain for anyone wanting to approach for the obvious reasons. Sadly he is at his weakest when leaving the sanctuary of his sentry to push forward, and should rarely attempt to do so unless he is certain he won't meet opposition before he got another sentry up.
Call me slow, but it was only when staring at the character selection screen for a while that this actually dawned on me; The heavy is positioned between these two classes. In fact he's in the middle of all the classes.
The Heavy is a hybrid class. He is not a jack-of-all-trades, in this sense the Soldier is superior. But the Heavy incorporates two very important aspects: Area denial and high DPS (damage per second). In essence he is a sentry gun, and has the ability to completely lock down a room if he is positioned right. But unlike sentries he is mobile. So if you notice that your position is getting too hot to handle you can relocate and avoid destruction.
Some think that a Heavy is only half a man if he hasn't got a medic trailing after him, and this is true for over-offensive Heavies. It's also true that the Heavy is at his most spectacular when he is plowing into the defending force with his minigun in hand and medic on tow -- but what about when there aren't any medics around? When you're dedicated to the Heavy class, you'll play several games without any medic at all. You'll have to play smart and draw upon other qualities.
Positioning becomes ever more important as a solo Heavy. You can't afford to be too reckless with the 300 health points you have, so you need to accept having to retreat a lot. If you have experience playing a Pyro you'll know that retreating out of harms way is the key to success. You learn to appreciate the "Live to fight another day" mentality. This is very important for the Heavy, as waiting for respawn and having to walk all the way to the front line with the slowest character in the game greatly diminishes your overall damage output. Also you'll be familiar with ambushes, which is something the Heavy is quite decent at -- even though he is a huge fucker and has got a gun that is noisy as hell when spinning.
Even though you'll be retreating a lot, you'll also have ample opportunity to advance in your enemy's direction even though you don't neccessarily know what you're going to run into. The key here is your shotgun. You should never walk around with the minigun unless it's spinning. Even if you'll be able to enter fire-mode faster having already equipped the minigun, you are actually reducing your overall lethality. Having the option to pump a couple shotgun rounds into an incomming enemy before you retreat behind a corner makes all the difference when you finally decide to take the minigun out.
Like I already stated how the Heavy has Area Denial capabilities, almost half of this is derived from the enemy believing that a room is that much more dangerous to enter if a Heavy is in it spewing his bullets. With you running away from them with your puny shotgun, you are purposefully withholding your ability to instill fear in them, and more often than not the enemy will become over-confident. Perhaps he'll even think you're out of ammo for your minigun. In either case he'll be in for a nasty surprise as you're just done with revving up when he turns the corner.
This tactic is laughably effective against most classes except the Soldier and the Demoman. But even they can't survive a point-blanc encounter for long against you -- even less so when you've chipped off some of their health with your shotgun.
You should strive to maintain your advantages as a Heavy, and health is one of the major ones. 300 HP at all times! Therefore it's essential you know the position of every health pack in every map, and possibly get a gut feel for when they respawn so you can better plan your retreats and advances.
Since you're slow you should limit your fighting to areas where health packs spawn. Even hanging around a small-HP spawn gives you a nice edge over your opponents. With the degree at which you can hold your ground proportionally rising in accordance with what HP-spawn you're at, the heavy can at times out-perform heavies that have a medic on them.
Having started out trying all the classes in the game, and then mostly favoring the Soldier, I'm beginning to use the Heavy more and more. Everyone and their grandmother can rocketjump about and spam rockets, but mastering the heavy takes patience -- and you feel less dirty when critting with a heavy than with a Soldier's rockets. While I do think the Soldier is perhaps a bit too good (along with the demoman), rocking with the Heavy is just as fun even if you might not top the scoreboard when you feel you did amazingly well.