Bound by Flame

This achievement is for setting enemies on fire, not for using fire on them. Big difference.

Enemies, when set aflame, will have fire visibly on their bodies for a few seconds. They are usually stunned or interrupted as well rendering them incapable of attacking briefly.

Similar to the poison effect you will have a chance percentage of "Set on Fire" with certain weapons or skills.
I have yet to find any weapon that comes with the Set on Fire skill attached to it, so that means you will have to either upgrade your Pyromancy Skill Tree or Craft a weapon/armor attachment with an ability that increase your chances.

You can check your Set on Fire percentage in the your menu (hit the BACK button). Scroll over (RT/LT) to the ATTRIBUTES section, then scroll (RB/LB) to the far right which is the NEGATIVE EFFECTS subsection.
From here you can check to see all your skills and abilities for your %chance of Set on Fire. Be sure to use the skills that have a positive % frequently.

There are several attachments for weapons and armor that can add the "Set on Fire" ability for either your main weapon or other Pyromancer skills.

Upgrading your Pyromancer skill tree is going to be your best option though, fire damage is very helpful against all enemies so you should try to upgrade these skills anyway.

Conflagration - is going to be a must have. The first point in this skill adds a 20% chance to Set on Fire for the Orb of Fire and Burning Weapon skills.
The third point in this skill will make Critical Hits automatically set enemies aflame.

Avenging Flames - is another skill but is less helpful. The third point in this skill adds a 30% chance of setting opponents aflame when they strike you. - Not as helpful

Fire Spirit - doesn't directly contribute to your percentage but it will allow the spirit to shoot fire orbs with the same properties that are on yours. - So these will have the same % chance and auto critical bonuses.

Getting these skills will use up a good chunk or you skill points so either go for this with your pyromacer playthrough, or a playthrough were you are not limited to 1 skill tree.