I suggest you start your knight with the Soul Master trait, as it makes completing Dog Princess's story a little easier. I'm going to walk you through everything in detail because I don't know how much of the game you played and I want to be as clear as possible, so sorry if some of this ends up being unnecessary to you; I just want to make sure even a new player knows what to do here.
Dog Princess is located far to the south. Go through the wendigo tunnel, through Clocktown, and then go south past the Rabbit Vineyard. You'll see a tower that appears somewhat dilapidated on the world map. Go to that. The first time you enter the area, there will be nobody there. Walk up to the door and interact with it. It will be locked. When you leave, a scene will play. You can return immediately after to discover two goblins and two bandits. Kill them and leave again; another scene should play. Come back one more time to discover the door open. Enter and go up to the second floor. There's a letter on a table. Read it and then grab the key that appears afterwards. Open the door and go up to Dog Princess. An NSFW scene will play and then you'll wake up the next day. Talk to Dog Princess and wake her up.
After talking to her, she will lock up any other princesses in your party and you'll be given the option to commit to her. If you choose to do so, you'll start her route. You'll be teleported downstairs. You may have already noticed that there is a chest down there. Open it up to get your second Filled Princess Soul Urn.
After you leave the tower, put some points into Dog Princess's love stat. (I'm not sure how many she needs to progress. I go with 5, I know that's enough, but she might not need that many.). Talk to her at camp and you will have a scene where you ask her where she came from and why she was locked in the tower. From here on out, you'll need
IMPORTANT: Dog Princess is a lot less tolerant of infidelity than most other princesses. Do not sleep around with other NPCs with her in your party. She will kill you if you set up camp within 7 days after your last act of infidelity. Also, do not drop her off at the Tavern. It seems to become inaccessible after you pick her back up. Not sure if a bug or an intended consequence. One last thing: always sleep with her every time you set up camp. This shouldn't be hard, since you don't exactly have another princess to choose from at this point.
You need 3 Filled Princess Soul Urns to progress to the next stage. I assume you started as a soul master and that you took the free one from Dog Princess's Tower, so you'll still need to find one more. Buy an empty princess soul urn from Merchant Princess. She will sell them at any location except for the tavern. Inconvenient, I know. You can fill it by "rescuing" a princess under siege, forcing her to abdicate, and then choosing the option to kill her. If there is no war going on or if you don't want to waste time battling until a siege becomes available, you can also kill Wendigo Princess. Equip a blunt weapon, head into the tunnels, and go to the far south near where it exits out to Clocktown. Smash the rocks in your path until you reach a doorway in the cave. Here you can fight and kill Wendigo Princess, but you'll have to exit the room and re-enter it to collect her soul.
Once you have the third soul, find Dragon Princess I's den. You know where Dog Princess Tower is? Go around the mountains and you should find it. It's a dragon's head on the overworld map that's near the scorched landscape. Can't miss it. Go to the second floor, then go into the other doorway, which will lead to a basement of sorts. Place the urns on the three pedestals and go through the portal. You'll wake up in the Dragon Queen's Palace. Go up the stairs and watch the scene play out. After that, you'll need to go back down the stairs and sneak around some guards in order to reach Dusk Princess (she's that one impish lady that accosts you and Kobold Princess after you two set up camp for the first time). Talk to her and watch another scene play out. After this, you'll be teleported out of the Dragon's Palace to a place near Dragon Princess I's den, but Dog Princess won't be with you.
Go back to her tower one more time all the way to the place where you she first slept with you. On the bedroll there should be a new item. Grab it to discover it's her pendant. Put the pendant on and don't take it off for the rest of the route. You don't need it right now but it's necessary to complete the route later.
At this point, you're nearly done. Go back to the tavern and, if you haven't already, do quests with Knights Inc. until you hit level 3 with the guild. Then, take the red quest. Drop any other members off at the tavern and go to the Fairy Forest. It's around the mountains where Golem Princess's Kingdom is. Go up until you see a mysterious hooded figure next to a save crystal thingy. It's Dusk Princess. Save the game and then let her warp you to Dog Princess's town. If you haven't equipped the pendant by now, make to sure to at least put it on before you do anything in the town. Again, keep it on for the entirety of this quest. It's important both for story context and actually completing the quest.
After warping, leave the house and go to the house to the left. Move the chest on the left wall and go downstairs to interact with the coffin. After the strange scene plays out, leave the left house and go to the well in the center of the village. Interact with it and you will see that there is something down there. You can't reach it from the well, though. In any case, check the upper left house in the village. You should notice something shining on the shelf. Interact with it and a fairy named Brownie will give you half of a pendant. Then, leave the house and go north to discover a pit you can climb down from. Go down and grab the thing that was at the bottom of the well: the other half of the pendant. Go back up and head to the palace. Weird skull monster things will attack you. Kill or ignore them and interact with the door until the pendant is in place. Go inside, go to the second floor, and head into the throne room. Read Dog Princess's note from when you got the pendant (I believe it's "Mysterious Note" in your inventory but that's going off of memory). Grab the Queen's scale from behind the curtain and head back outside.
Go back to where Dusk Princess is and "proceed with the next step". However, don't tell her where you got the scale from. Ask her what's she's going to do with it instead. She will dodge the question, then begin to loop a conversation until either you tell her where you got it or attack her. If you tell her where you got it, she will teleport you back and kill you. That's the bad ending. Attacking her will "break" the illusion and you'll learn that you were in the forest this whole time. Dusk Princess is not a boss. She will die in one hit no matter what your level is or what weapon you use. After she dies, go back south to discover Dog Princess waiting for you. You two will flee the kingdom and one last scene will play, having earned your happy ending. Finish the playthrough and the achievement should pop up.
Man, that was long-winded.