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So, first things first: Queen Victoria's name is actually Alexandrina Victoria. (British monarchs get to choose their regnal name and she opted to drop Alexandrina.) Now on to the murkier and more interesting topic: how'd she get to be the queen of England?
In 1817, George III was king and his eldest son, George IV, was Prince Regent. The Prince Regent's wife, Caroline, had been estranged from him for years. Their only child, Princess Charlotte of Wales, then died in childbirth that same year, leaving him without a legitimate heir.
This meant that his brother, Prince Frederick, the Duke of York and Albany, became his heir. However, Frederick and his wife also didn't have any children.
Several of George III's other children hurried to get married and produce heirs on the increasingly likely-chance that the succession came to them. William IV and Prince Edward, Duke of Kent and Strathern (George III's third and fifth children), both were married in 1818. Alexandrina Victoria was born the following year.
Both Prince Edward (Victoria's father) and George III (Victoria's grandfather) died in 1820. The Prince Regent became king.
In 1827, Prince Frederick, George IV's younger brother (George III's second child) and heir, died. Since Frederick never did have any children, their next brother, William (George III's third child) became George IV's new heir.
George IV died in 1830 and William IV became king. He was in his 60s by this point. William had several illegitimate children prior to his marriage to Princess Adelaide, but unfortunately they did not have any surviving children. Princess Charlotte (George III's fourth child) had died in 1828 without any living children, so as Prince Edward's daughter, Victoria became heiress presumptive to the crown.
And if you're still wondering about the old duke with the scar from the show? That's Ernest Augustus, George III's eighth child, making him Victoria's uncle. If it weren't for her, he would have been King of England. (He did, however, became King of Hanover* on William's death since they had a law preventing women from inheriting the throne.)
*This could be its own topic entirely, but essentially Hanover was a short-lived kingdom in the Prussia/Germany area created by the Congress of Vienna and given to George III.