After getting the White Crystal and some Holy Water, you can go to Berkeley Mansion to get the first piece of Dracula.
Get an Oak Stake
In order to get the piece of Dracula from the mansion, you are going to need an Oak Stake. Luckily, there is an Gypsy in the mansion who will sell you an Oak Stake for 50 hearts.
To reach the Gypsy, first you have to equip the White Crystal so you can see the moving platform at the start of Berkeley Mansion. Once the platform is visible, jump onto it, then when the platform is almost at the top, jump from it to the higher platform on the right.
Try killing the Knight with the whip or other weapons that you have with you, then jump onto the platform where the Knight was. Then go to the right, killing a couple of skeletons on the way, until you find some stairs going up. Go up those stairs, but be careful of the Slimes that are up here. If you got the Sacred Flame in Dabi's Path earlier, it works well on the Slimes.
Go to the left and you will encounter a Gargoyle. It spits two flames at an angle, and it jumps randomly. Kill it and go to the left. Jump up the small platforms, killing Skeletons as you go. Then kill the Knight to the left, and jump onto the platform where the Knight was. Go up the stairs there.
Go to the right and kill the Skeletons that you encounter. When you see a higher platform to the right, DON'T JUMP! The two blocks in front of the column are fake! Be sure to jump carefully so that you land on the block that is immediately to the right of the column.
Then keep going to the right, killing more Skeletons. After going to the right, you reach a new area. In this new area, there are Skeletons that throw bones at you. Be sure not to touch the flame that the bone creates, because it will hurt you.
Keep going right in this new area until you fall down from the ledge. Then go down the stairs where you landed. Go to the left, then jump up onto the raised platform on the left and go up the stairs.
Go to the left, jumping over the spikes, and you will find the Gypsy. Buy the Oak Stake from him for 50 hearts. If you don't have enough, just fight skeletons and other monsters here in the mansion until you have enough.
Get the Piece of Dracula
When you have an Oak Stake, go to the right from the Gypsy, jumping over the spikes. Go down the stairs, then go to the left and go down these stairs. There is a third set of stairs. Go down them. Now go to the right. Beware! There are some fake blocks in front of the second column! If you fall, you are going to have to go to the left until you reach a wall, then go up the two sets of stairs, and then go through almost the entire mansion again. So be sure to jump over those two fake blocks in front of that second column.
After jumping over the fake blocks, go to the right and go up the stairs, then go to the right again. Jump up to the platform where you see a glowing orb. Equip the Oak Stake and throw it at the glowing orb, then jump onto the bag to get Dracula's Rib. If you equip it, you will hold a shield in front of yourself when you are jumping or standing still, but not when you are walking.
Leave the Mansion
Now that you "prossess" Dracula's Rib, go to the left. To get out of the mansion more quickly, you should deliberately fall through those fake blocks that I warned you about earlier. Just walk left without any jumping, and you will fall when you reach those fake blocks. After you fall, just go to the left until you reach a wall, then go up one flight of stairs. From there, go to the left, jump down to the lower platform, then jump to the platform on the left and fall through the hole in the floor. Then you can go to the left to leave the mansion.