Aesthetically-speaking, it obvious that Bloodborne takes a great deal of inspiration from the Gothic. But what of its themes? Well, let's explore the connection between Bloodborne and its Gothic influences a bit:
Religion is Corrupt
In many Gothic novels--especially those written by 18th century Britons--the Catholic Church is portrayed as a corrupt institution. For example, Matthew Lewis's The Monk makes the case that the Catholic faith's culture of sexual repression allows the titular monk Ambrosio to be easily seduced by the Devil's ploys, as well as indicting the power, wealth, and pageantry of the Church as evidence of its focus on worldly matters instead of spiritual concerns. Other Gothic fictions, such as Poe's "The Pit and the Pendulum" and Ann Radcliffe's The Italian, dramatize the Church's excesses as sadism that operates under the guise of religious correction.
In Bloodborne, your character has come to Yharnam to be cured of an unnamed disease by the blood ministration of the Healing Church. However, as you progress through the game you realize that the Healing Church doesn't offer blood ministration simply out of altruism; instead, the Healing Church is using blood ministration to control the populace while using that same population as lab rats for its various experiments with otherworldly blood. Notice that if you send any survivors you encounter to Iosefka's clinic, they will be experimented upon and that the impostor Iosefka wears the garb of a Healing Church agent. Furthermore, the Healing Church is shown to be not just responsible for the beastly transformation of the populace; it is also involved in a cover-up of its responsibility. Ultimately, the Church Hunters are actively disposing of the victims of Church's experiments, and Old Yharnum was burned in an attempt to destroy evidence of the truth about blood ministration.
Man's Nature is Bestial
The results of the Healing Church's blood ministrations reveal another Gothic theme: man is not so far removed from base beasts as we might like to believe. In Rudyard Kipling's "The Mark of the Beast," European colonists in India discover that the heart of a predator beats within their breasts, despite their ideological belief that as white men they represent the apex of civilization. When one of their number is transformed into a werewolf-like creature, it is revealed that European culture and the norms of "civilization" are simply a veneer that hides the savagery of the human condition. Similar transformations are laced throughout Bloodborne: the animal-like men and women of Yharnam created by the Healing Church's blood ministration are further evidence that it's possible for civilized people to devolve into ravening beasts.
Oh, and it's already happened to you as well. Take a look at your hand when you perform a visceral attack; the reason why your hand is monstrous and claw-like is because you too have been tainted by blood ministration and have begun to backslide into a bestial state. Father Gascoigne's plot line isn't just the horrifying tale of a man who gives in to bloodlust and kills his wife--he's a cautionary tale about your continuing degeneration into something both more and less than human.
Science and the Supernatural Are Not Separate
I think people sometimes get the wrong impression about the existence of a well-tended dichotomy between science and the supernatural in Gothic fiction. While in some stories there is a binary opposition between science as a manifestation of reason and progress and supernaturalism as an overflow of passion and superstition, it's more often the case that science and the supernatural work in tandem. Note that in Shelley's Frankenstein Victor's education in the discipline of natural sciences combines with his research into discredited alchemical theories to produce the secret of creating life from inanimate matter. In Machen's The Great God Pan, neurosurgery is used according to mystical ideas to pierce the boundary between the material world and a spiritual plane of existence. In Stoker's Dracula, the only way the heroes manage to defeat the vampire is by utilizing the latest technology (blood transfusions, steam conveyances, the telegram, phonograph recording) alongside Van Helsing's knowledge of ancient folklore and superstition.
The obviously supernatural horrors of Bloodborne are all related to scientific inquiry and experimentation. The discoveries about the alien Great Ones made by the scholars of Byrgenwerth are pursued with scientific rigor; the Healing Church's blood ministrations take a magical substance and apply it via medical science; the inventions of the various hunter's workshops meld supernatural effects (such as the sparks of the darkbeasts) with weapon technology to create new implements of battle (such as the blue-sparking tonitrus). Bloodborne's terrors sit firmly at the junction of science and the fantastical.
Abjection is Everywhere
The importance of blood is a given in a game named Bloodborne, but it's interesting how often blood is particularly associated with women in the game.
As far as I've discovered, only women give you special vials of blood. Iosefka, Adella the nun, Arianna the prostitute each offer you a special kind of blood vial, and the last two give you blood that is explicitly their own. (Also note that both Adella's and Arianna's behavior changes with the rise of the Blood Moon.)
Queen Annalise is a vampiric "vileblood"; swearing an oath to her involves drinking her blood, and she will then send you in search of blood dregs to bring back to her. Lady Maria of the Astral Clocktower makes use of a weapon that requires her to shed her own blood; as the fight with her progresses, her attacks become accompanied by waves of damaging blood residue.
The reason why there are so many close associations of blood with women in Bloodborne is to reinforce the idea of abjection. As Julia Kristeva asserts in Powers of Horror, abjection signifies being “cast-out” or marginalized, and the descriptive tropes that most often mark this state are blood and other "disgusting" bodily fluids. (Note that Lady Maria also seems to have something to do with the experiments with brain fluid in the Old Hunters DLC.) When women are depicted in terms of the horror that often attends Gothic descriptions of menstruation or childbirth, they're being portrayed in terms of the abject; because the biological realities of being a woman are made monstrous in these stories, the very state of being a woman becomes associated with disgust, revulsion, and excrescence.
That Bloodborne couples its female characters with images of blood makes sense when you realize that the central premise of the game is one of horrific childbirth. After all, Queen Yarnam is both associated with blood (the front of her white dress is stained with her blood after she gives birth to a child given to her by the Great Ones and when you fight her she turns into an erupting fountain of tainted blood) and with a distorted, horrifying image of grotesque motherhood. Arianna's story line reinforces this connection as well; not only is she associated with blood in two ways (she gives you blood vials and her blood is related to the vileblood of Cainhurst), she is also fated to give birth to an inhuman monstrosity. Also, look at the blood dregs you collect for Annalise...they're some sort of bloody sperm. No wonder they're said to allow Annalise to give birth to a "child of blood."
It's called the Nightmare of Mensis for a reason.
Have you ever attacked the Plain Doll in the Hunter's Dream? Notice that she bleeds too. Also note that she will care for you as a "child" if you complete the "good ending." Add that to the speculation above about women, blood, and childbirth, if you dare.
But let's also dwell on what the Doll adds in terms of Gothic conventions. The Doll is uncanny. She looks human, but it's clear from her dialog and strange, stilted mannerisms that she is an automaton of some kind given "life" by the Hunter's Dream. As such, she represents Freud's ideas of the unheimlich; since she blurs the line between animate life and inanimate object, she becomes an alien, disquieting figure. The Doll functions a lot like the statue in Thomas Hardy's "Barbara of the House of Grebe": she becomes the uncanny, material substitute for the kind of safety that is craved in its real, human absence. Of course, you'll eventually realize that despite not being human she's easily one of the more safe and secure characters you'll meet in the game. Except...take a look at the color of her blood again. What other creatures have that color of blood in the game? Best not think on that too much, good hunter.
The Present is Haunted by the Past
When Hamlet's father tells him that "the time is out of joint," he essentially codified the most important convention of Gothic literature: something horrible happened in the past and the reverberations from that event continue to haunt the present. The idea of haunting is central to the first Gothic novel, Horace Walpole's The Castle of Otranto: the usurpation of the throne generations ago has resulted in both a tyrant acting as prince and the intrusion of specters from the past into the present. The fictions that followed Walpole's Gothic mode applied that convention as a tenet of faith.
Your character in Bloodborne similarly finds themselves encountering the troubling echoes of the misdeeds of the past. Why else would your experience points in this game be called "blood echoes"? Of course, you act as a hunter of monsters, dispatching both the transformed residents of Yharnam and the Great Ones who have been beckoned from across the cosmos, but you're forced into that role because of the actions taken by the Healing Church, the Byrgenwerth scholars, and the various hunters of the past generations. In fact, once the night of the hunt begins it's clear that time has stopped working in a linear sense--you meet characters who are long-dead or are otherwise trapped in the Hunter's Dream. One of the endings lets the break the cycle of repeating the past, but the other two keep you firmly within the cycle of disjointed time. The great irony of Bloodborne is that descending into the past to solve the riddle of what went wrong literally makes you part of the problem: if horrors were unleashed by the scholars of Byrgenwerth and the tomb prospectors when they explored the ruins of the ancient Pthumerian civilization, you merely extend the circular nightmare by delving into the history of those explorations.