No, Chalice dungeons are completely optional. However, they will reward you with lots of Blood Echoes you can use to level up, blood gems to improve your weapons and insight for killing bosses. The higher level dungeons have blood gems that are more powerful than those you can find in the main game, so playing the dungeons can make the end game and NG+ a lot easier. If you're ever stuck because an area or boss is too difficult it can be a good idea to visit the dungeons to improve your character, but you can beat the game just as well without doing so.
In case you care about Playstation trophies, you'll need to visit the dungeons to get one for beating the final dungeon boss and to find a weapon that can only be found in the dungeons to get the trophy for acquiring all weapons.
Regarding where you need to go after you beat Vicar Amelia: It's been a while since I last played the game, but from memory I think you're stuck at the same part as I was and the solution is probably:
Completely spoiler free: You need to find a door in the Cathedral Ward area not too far from the Vicar Amelia boss room where some giant enemies are around.
Directions and mini spoiler maybe:
After killing Vicar Amelia you need to touch the skull on the altar in the boss room, which triggers a cutscene (you probably did so after the fight naturally). In the cutscene you see an old man talk about the "old blood". This cutscene reveals the password for a door. To find the door, from the Vicar Amelia lamp leave the boss room and go straight all the way down the steps until you are in a big round area with tomb stones and some giant enemies. From the gate you entered that area stay right and go through the first gate past a giant and follow the way down the steps.