Three things.
1. Get better gear. DD is a very gear heavy game. Gear matters even more than how you build your character imo. That doesn't count for your pawn though. That's a whole other issue.
2. Make sure your character is set up correctly and has good augments.
3. Hire a full party that has good inclinations(hint, avoid Guardian).
Also, were you attacking the heart? Always aim for the glowing spot on his chest. Since you're an Assassin you can jump up to it, re-equip your daggers to a sword, then do Dire Gouge. You should do massive damage to him with that skill if you're playing Vanilla DD. If you don't have that skill Hundred Kisses also does the job. Just make sure to take Liquid Vim or mushrooms. If you attack at the lower right(your right, not his) he won't be able to grab you off of him either. Nearly every large monster in DD has a "safe spot" where they can't attack you. If you want to play it safe just spam Fivefold into his heart, it will eventually stun him and he'll fall on the ground for about 20 seconds. I will say this though, if he starts flying you will have to use the various balista around the area and shoot him when he comes swooping down. If he's on his perch shoot arrows at him, that's all you can really do. Your pawn won't be too useful in this fight though. In vanilla DD warriors aren't too great against the bosses. I would recommend switching him to fighter after you get the Clout augment for him.
About the Gorechimera, attack the goat first. The goat is what takes the most damage, you can also kill the snake off to stun it for a short while(about 10-15s). Just remember to take the goat out as quick as you can. If you leave it alone the goat can heal it for 2 bars of health. They can also cast High Exequy, you'll notice it if a thick red circle appears around you or your pawn. It kills anything that is within the circle when the spell is done being cast. So taking out the goat is always a priority when fighting them. Once the goat is dead it's just a matter of killing the lion and the regrowing snake.
EDIT: Forgot to mention, The Dragon is weak to dark damage. If you have a way of dealing dark damage it will really help. He resists holy so don't use that. If you got the kunai daggers in the chest near the skeleton lord during the quest before this you'll have access to dark damage. It's the only time in DD where I can honestly say dark damage is useful.