Mary was foremost a staunch Catholic. Even when it might have aided her to convert to Protestantism to keep hold of the Scottish throne, she retained her faith. When she was executed, she wore a red dress to symbolize that in Catholic style she was a martyr. This is important to your question.

Mary was a young child when she inherited the Scottish throne, and she was sent to the French court far younger than she would have been in ordinary circumstances. Mary was raised alongside Catherine de Medici's royal children as though she were a member of the French household, rather than so much a separate entity and queen in her own right.

Mary was betrothed to the dauphin, or heir apparent, to the French throne. From her childhood Mary was raised as a member of the French court while being told she was potentially a right and true heir by God for the English throne. Edward VI died without issue, Mary I didn't live long and died without issue and Mary I had almost been denied the throne. Mary was the great-niece of Henry VIII through a legitimate marriage. Elizabeth I had always been on shakey ground in terms of legitimacy, as Catholics in particular didn't recognize her claim to the throne or Anne's marriage to Henry. Elizabeth was a bastard in Catholic eyes, and outside of the sanctity of the church was a Protestant heathen to boot.

Henry II envisioned that Mary of Scots and his son Francis would bear a son who would inherit Scotland, France, England and become a superpower. And although he seemed to care less about the religious aspect, it would please the Pope and Catholic people to have Scotland and England brought back to the church.

Mary was raised in her upbringing and religion to believe Elizabeth was a heathen bastard and that her own blood was the true claim. Thrones were granted by divine bloodline. The idea that monarchy is a calling rulers are brought to and chosen for by God. This is why they are crowned by the holy office and annointed in holy oil. This is part of why it was difficult to justify executing Mary of Scots later. She was, in medieval eyes, God's annointed queen.

Mary chose to style her coat of arms in the fashion to declare herself the English queen. And there was a dinner where at least one Englishman [possibly an ambassador, I don't recall offhand] came to see Mary and she was loudly and purposefully announced on arrival as the Queen of England. This is a big slap in the face to Elizabeth and a big declaration that she didn't recognize Elizabeth's claim or authority.

She also wasn't terribly fond of Scotland, all things considered. She was raised in the French court which considered itself the pinnacle of culture and elegance. The unruly, 'backwater' Scots with their cold, wet country seemed a punishment to Mary who was depressed when she was made to leave France to travel there. The English throne was preferable to the Scottish one to her, and she married Henry Darnley to bolster her claim as Darnley had a bit of the royal blood in him through his grandmother, Henry VIII's sister Margaret. Her own son James skipped town from Scotland and spent most of his time in England as soon as he was to be crowned.

In religion, the way she was raised, her own desires, and the indeed the pressures of schemers like Henry II and the church, it was ingrained in her to want and seek the English throne. And also to believe it was her 'duty' to God to save England from Elizabeth, rather conveniently for her ambitions.