The first thing you are going to want to do before fighting the Thermophile is get the following Knowledge items discovered:

• The Meteor Sword
  
• Thick Fur Clothes

Once acquiring both of these, make sure to stock up on Pikken Meat (make sure to cook it!) prior to fighting the boss, as it will help you stay at full health leading up to the fight. If you can, get the Tinctures you obtain from completing the Playing Mouse quest and save them for the fight to heal.

Before you fight the fiery wolf, you must go into the fiery hill. Once inside, you will want to know what enemies will show up inside, so let me give you a rundown on each:
The most annoying of the 3 enemies. It takes several hits to defeat them, and being NEAR them raises your Heat meter. Letting it hit maximum sets you on fire, and by then the tick damage and constant stabbing from it's obsidian knife is sure to leave you lying on the floor with a good 50 stab wounds and 3rd degree burns. To counter these, stay away from lava pits and slash wildly. The knockback will send them towards one of either your left-hand or right hand side of your character. Change your attack angle accordingly or they will be out of your attack range, letting them get a few hits in.
You've seen these guys before, and they are quite easy to deal with. The Meteor Sword kills them in 1-2 hits, so make sure to kill them for some easy EXP.
These guys are the middle ground, while they cannot be damaged by physical attacks (they are highly resistant to them) using a special attack will destroy them in 1-3 hits on average. That said, they are very easy to take care of, but do not let them get into melee range. They deal extremely high damage (especially due to your lack of physical defenses, it's a one-two shot hit) at close range but are extremely slow to float towards you.

A simple guide to get through the volcano is to bring a pickaxe and climb up anything you can easily jump over to reach the boss area in about 3-5 minutes give or take any complications.

Now, to actually fight the beast!

The most important thing to do is to jump over it's Meteor attack, it will send out a wave of shrapnel that you can easily jump over with good timing. If you're a fox, abuse your Foxfire special attack to deal extremely high damage to the Thermophile early on into the fight. If you are anything else, you will have to resort to baiting out his Fire Breath attack. Be wary, however, because it has decievingly far range and obscene knockback. Due to the large amount of lava pools, you could take an unexpected dip that will lead to your demise.

Speaking of death, one of the most lethal things this boss can dish out is summoning Imps. While the Thermophile's attacks are easy to dodge in a typical situation, the Imps can be summoned in large groups, which can turn the tables of a fight extremely quickly. The best thing to do during this situation is to make distance between the Thermophile and-during one of it's attacks (Preferably Fire Breath, but Meteor works if you are well out of the way of the shrapnel)-quickly slice down the Imps before they can deal any damage.

• When the Thermophile uses it's Meteor attack, standing directly to either side of it's body will render you invulnerable to any of the Meteor's shrapnel, letting you get a lot of hits in.
  
• Jumping toward the Thermophile instead of jumping in place to dodge the Meteor attack gives you more time to react to the Shrapnel and jump over it, if you are caught directly in the line of fire of said attack.
  
• Don't give up, this fight will ruin the unprepared, especially if they lack healing items. Be careful and stay strong.