For the entire battle, Furnace has 3 main attacks you need to look out for:

Spinning: Furnace's eyes light up and have smoke emit from them, with a longer scream. During this attack stay as far away as possible and take the time to heal, if you are cornered dodge around the edge of the blade.
Stabbing: A shorter scream just move either side of him and use explosion for extra damage.
Dash Spinning: A longer scream just move to Furnace's left (the side that the blade isn't on), and use eplosion for extra damage.

Once reaching 10000 health, Smelter will spawn, a bladeless Furnace that does the same attacks without the weapon.
For dodging dodge through Smelter as he lacks a weapon.
Kill Smelter first for the second achievement.

For non achievement-ers
Target Furnace until he dies, as it is a pain having 2 bosses in the arena. Use the same pattern as the first phase just with 2 bosses.

This goes for both bosses, upon reaching 5000 health, they will turn golden and gain faster attacks, as in no delay between attacks, and primarily choosing the stab attack.

This is only if you kill furnace first.

Once Furnace dies, smelter gains the sword and attacks faster, but is basically the same as the first phase.