Tainted Grail: Conquest

This achievement is much easier than you might think. The most important thing to know is that you can use potions to heal. If you go into your inventory, you can use them there if you need/want to heal between fights. You can also use Meat to heal you inside of combat, hunting at the Hunting Grounds should give you a bunch, they heal 5HP each.

This is technically possible with any class, though some make better choices than others. I earned this using the Blood Mage class, though I have also done no heal runs using the Zealot and Sentinel, and could also recommed using the Summoner, Berserker and Pathfinder classes.

General Tips:
Health Fountains always have one partial heal for free and one full heal available, if you stay and beat a combat
All Mother's Altars have a small partial heal as an option
Take Max HP buffs whenever they are offered
Don't take any choices that lower your max HP
Stan Runestones increases Max HP by 30/50/90 when used in the Armour slot (level 1,2 and 3 respectively, each tier is 3 of the one below)
Calc Runestones heal you for 5/8/15 HP after every combat in the Armour slot
Lor Runestones allow for 10/18/30% Lifesteal (get x% of damage as HP) for the first Hit every turn when in the Weapon slot
Or Runestones heal you for 5/8/15% of your max HP after every combat in the Armour slot
The Bard has a White Card that lets you heal 10HP once per combat, buy it.
Items can help you out against very tough opponents. Acid Flasks, Acid Bombs, Overcharged Runestones, Consecrated Oil and Smoke Bombs are the best choices

If Zealot/Sentinel:
Armour gives you Barrier, Barrier is life. Always take Armour buffs , unless max HP is an option
Abuse your Blue cards every turn if you need to, you should be able to take almost no damage every combat

If Pathfinder:
Abusing your Block generating cards is the intended way to play the class
Get the Passive Skills Agility and Natural Block
If a card gives you Block and there isn't a great damage dealer, take it.
Your Ability card, Insight, doubles the Block gained from a card if played directly after
To fight effectively, you should be able to take no damage each turn.
You only lose Focus if you lose HP. A combination of Block and Barrier is godly

If Berserker:
You will take damage, don't bother Blocking unless its a really, really big hit
Only use your Ultimate at 8 charges or above (60% Lifesteal), 12 Charges only is 100%
The biggest danger is one massive attack near killing you, or many small combat where you don't get the chance to heal.
Take the Blood-Oath Mastery at Level 10, it gives you double damage on any attacks used in the same turn as your Ultimate, which means double the healing.
Skullcracker and Bloody Frenzy are amazing cards for healing, or just for bringing a lot of pain

If Blood Mage:
Don't summon more minions than you need
Blood Fae are useless, dump the summoning card
Wyrms best damage dealers, Abominations better for 3+ enemies only
Spam Activations when a minions is Level 4
You shouldn't need many Golems, but keep the card around just in case.
Minion buffs should be applied early and often
Keep a Barrier around the Blood Mage at all times

If Summoner:
Fae are near useless, dump the card
Always have a Barrier on the Summoner
Always have a Golem
Wyrms best damage dealers, Abominations better for 3+ enemies only
Spam activations and Promotions on your Wyrm unless you need the Energy for a new Golem or applying buffs
Summoner Minions don't go away unless they die, so buffs last a long time. Apply often
Use your Ultimate at 9 charges to get 3 Promotions
Mystical Blood greatly extends Golem lifespan, grab it when you see it.