It definitely is possible to win all 8 matches on hard. I've done it, and I've done it on Nightmare too. I've played on Nightmare far more than I've played on Hard, so I'll be better describing Nightmare, but it should apply to Hard as well.
First, it's important to understand the situation that you're in. In Tournament mode, your allies are always on normal difficulty, while your enemies are whatever difficulty you select. So in Hard mode, your allies are normal, and your enemies are hard. That is why it always feels like your allies are lagging behind you.
Also, you need to understand what an increased difficulty means: the computer is not at all smarter, they just cheat more. The computer will get extra health, mana, experience and money. If left to their own devices, they will eventually become nearly invincible. Oftentimes, one of your computer enemies will suddenly become incredibly powerful, seemingly out of nowhere, and remain powerful throughout the rest of the game.
In order to defeat the computers, you'll have to get many, many more kills than they do. In conquest mode, you'll probably want in the neighborhood of 20 kills by the time the game is done, though it's not uncommon to need more than that. In order to get this many kills, you need to engage enemy demigods at every opportunity, especially when it is one on one, and in order to defeat them, for most characters, you'll need to use your regular auto-attacks.
You may not want to neglect your abilities completely, but the computer players will have an incredibly high amount of health and health regeneration, and I've found that it's usually very difficult to do any significant amount of damage with most abilities. So you should focus on doing damage with your auto-attacks. On almost every character, you'll want to take the bottom row ability, which increases your damage or your minion damage. You'll want to be certain to purchase gloves and a favor item that increases your basic damage. And a general, you'll want to make sure you get the highest rank idols.
For example, with Queen of Thorns, I would take the shamblers ability, all of the shamblers upgrades, and all of the minion upgrades. I'd continue by purchasing +minion damage items and eventually artifacts that increase my damage. When fighting, I'd always make all of my minions attack the same target as me.
With Regulus, I'd take his Wings ability (that gives him bonus auto-attack damage), mines (mines are good for both leveling and killing CPU demigods), and the bottom-row ability. I'd purchase gloves, a mageslayer, and eventually Ashkindor. Snipe and Mark of the Betrayer are mostly useless against a nightmare CPU.
As for strategies killing demigods, just keep in mind that the CPU is not very smart, and will follow you, sometimes at great peril. Use your towers to your advantage: if you can't kill an enemy, see if you can't lure him into the towers. It's also a good idea to stick with your computer allies: they'll be pretty much helpless on their own because of their lower difficulty setting.
It's also a very good idea to try to keep constant control of the artifact shop (if you're not on a map where the shop is in the enemy base). Preventing the computers from getting artifacts is not incredibly difficult, because they won't necessarily prefer it over any other flag when they've got the money to buy an artifact, and it will actually work.
Also try to end the game early: try to buy catapultasauri as soon as you hit war rank 8, and once you've got mage slayer, go for the towers when there is no enemy demigod around. Ending the game early is important because the enemy demigods will eventually become far more powerful than you could ever hope to become. In Nightmare difficulty, Rook or Unclean Beast can easily become nearly invincible, and they're still rather dangerous in Hard. If you encounter one of these super-powerful computer opponents, you should try to kill them if they push too far into your base, but otherwise it's best to leave them to wander around while you attack the other demigods and push toward the enemy citadel.
Hopefully some of this advice helped...let me know.
- Jaxian