Consider the use of momento mori in the play. When Hamlet is talking to the grave diggers and the dig up the skull which Hamlet then picks up and addresses. This is an example of momento mori and it was used to cause characters (and audiences) to consider their own mortality. It became very common in plays and literature, but Shakespeare was the first person to name the memento mori "Alas poor Yorick. I knew him well."
You have gotten an excellent summary of the Gothic elements in Hamlet. The area I want to address is your request only for such elements that have a direct impact on the development of the story. I'd argue all the above-mentioned are Gothic features which all add to the story's development. Without the Ghost, Hamlet would have taken no action; without the elements of hell and damnation, Hamlet would have killed Claudius earlier; without the "rotten"-ness of Denmark, it would not have been vulnerable to attack by young Fortinbras; if something didn't smell bad about his mother's hasty marriage to his uncle, Hamlet would not have been so ready to believe the Ghost...and the list goes on.
The Gothic writers also delved into the working of the subconscious or unconscious minds of the characters of their stories. This can also be described as a focus on the passion and irrationality of human behaviour, rather than the logical and rational. They created external environments, often, that reflected the internal crisis and chaos of the characters.
This examination of personal motive and extended investigation of passion and irrationality can definitely be seen in the actions of Hamlet himself. He is consumed with his own grief and melancholy. He does not do what a "rational" character in a revenge tragedy would do -- ie, murder the culprit and revenge his father's death. Rather he agonizes over his plight and his mother's marriage and even laments and bemoans the fact that he is unable to act.
Everything, it seems, that happens in the play holds potential for Hamlet to react to it with a sort of manic outburst of passion -- his encounter with the ghost, his "mad" playacting, the arrival of Rosenctanz and Guildenstern and the players, the enactment of the play itself, his encounters with Ophelia, his encounter with his mother, his murder of Polonius...the list could go on.
For me, the Gothic feature of Hamlet that most affects the development of the story is the behaviour of Hamlet himself, behaviour that emphasizes the irrational and passionate over the logical and rational.
The Gothic elements in Hamlet that drive the plot are the focus and preoccupation on:
1. Death: the word death is mentioned 39 times in the play. The action results from the death of the former king. Hamlet soliloquizes ad nauseam about killing Claudius and himself. Ophelia commits suicide, the gravediggers poke fun at death, and two men fight in an open grave.
2. Disease/decay/sickness: Part and parcel to the fixation with death is that which leads the characters to it. There's incest, poisoning, Yorick's skull, mental illness, and "something rotten in the state of Denmark."
3. Sense of mystery, horror, supernatural, unnatural: Besides the ghost scenes, there's a pervading sense of horror in the play, namely related to the unnatural relationships between Gertrude and Claudius (incest), Hamlet and Gertrude (Oedipus Complex), Ophelia and Hamlet (closet interview), and Ophelia and Polonius (pimp-prostitute relationship). All of these contribute to the violence, spying, paranoia, and psychological tension between characters.
4. Hell and demons: Hamlet is torn between believing his father's ghost is a demon or a spirit caught in Purgatory. He also is afraid of going to hell himself if he commits revenge against Claudius. Revenge is a pagan and Gothic (Medieval) practice that is revisited in this Christian-era play.
The most essential Gothic element is Shakespeare's use of the supernatural. The ghost's appearance to Hamlet in the form of his father clearly creates a sense of foreboding and urgency to the play. Shakespeare's audience would have believed in ghosts to some degree and would certainly have understood the convention that if a ghost appears, something is not right in the world. The folklore of ghosts suggested that "true" ghosts came to provide information or premonitions or to finish unfinished business, but that ghosts could also be just the devil in the guise of a loved one. Those types of ghosts came to cause harm. Hamlet must deal with this supernatural phenonemon and determine if the ghost is a "true" ghost and if his words can be trusted -- he therefore spends the next two acts of the play determining Claudius' guilt in the death of his father.