Blaine Anderson knew he shouldn't exist. He knew the time lines of his parents had never matched the way they were supposed to. They never should have been granted the time to conceive, but the universe granted it to them anyway. He was proof that the time stream could become mixed up beyond all hope and yield some sort of positive result. He was proud of who he was, proud to be like his parents…proud to be a Time Lord. That's why, over the years, he gladly grew into the traits of his parents. He adopted his mother's fearlessness, common sense, and adventurous nature, but allowed himself to keep the childish attitude his father was known for, as well as his tendency to run and his name.
Taking the name his father was known by across the universe had always been the most important to him. He had always believed that his father, far more than his mother, had a hidden desire to find a way to live forever. His father had been "the last of his kind" for so long that Blaine thought it was the most proper way to respect him. After all, being "The Doctor" was the most spectacular thing he could do. It gave people all across the universe, all through the time stream, hope that someone will come to bring them from their darkest days.
Perhaps more than letting his parents live on (he never failed to mention his mother when given the chance; the Time Lady Archaeologist was something of a tale to tell), Blaine wanted to fill his parents' shoes. He wanted to be great like them, to be wise like them, to be known like them, to be heroes like them.
His wants, however, were not at all part of the reason he found himself attending a school for humans - loads of humans (he inherited his father's fascination with the human race) - in the middle of the state of Ohio, in the middle of the United States, in the beginning of the twenty-first century. Actually, he ended up in the outskirts of Westerville, Ohio because that's where the TARDIS (something his parents should have never let him see but did anyway) thought he had to be. Blaine's father once told him that he had the undeniable pleasure of speaking with the TARDIS, whom he had referred to as "Sexy," and that, although she didn't always bring him where he wanted to go, she always brought him where he needed to go. The entire concept fascinated Blaine, which is why he rarely had a specific destination, but rather let the TARDIS take him to the times and places she felt he needed to be.
What Blaine didn't understand right away was why he had landed where he did. Of course, he understood slightly more why she would want to bring him here when he felt that his constant traveling companion landed herself on a rift. Perhaps, he had thought upon first stepping into the open area of Westerville, she just stopped because she needed to replenish her energy. In the end, he had been wrong. It turned out that in addition to the convenience of a rift being there, that a desperately distraught married couple needed him.
It had seemed so small, so simple, to Blaine to be needed by Joseph and Beth Anderson. They had been without a son for several weeks and they needed the comfort of having someone around them again. And, being the son of his father, Blaine couldn't refuse; his soft spot for humans was too great. Joseph and Beth were the ones who gave him the name Blaine Anderson and paved the road for him to attend Dalton Academy after being in that horrific public school system for years (none of them had expected Blaine's attendance to proceed the way it did). He spent the years sharing intimate moments with the Andersons, adopting the formal, yet affectionate titles of "mother" and "father" for Joseph and Beth after they let Blaine open up to them. They understood when he had to get away, to go off to some distant time or place or people to make sure that the life he was born with was still there, so they let him go. For Joseph and Beth, Blaine wasn't a replacement for the son they had lost to an unexplained coma. To them, Blaine was like the second son they had always wanted but never had the courage to have. And for Blaine, Joseph and Beth were so much like the parents he remembered that he had to keep coming back to them.
The Andersons showed him music in ways he never would have dreamed to know before. They showed him piano and guitar and voice and orchestras and musicals and records. He had been taken aback by the brilliance of it all, involving himself deeply in singing. That's how he came to be where he was today: sixty years old, in his second regeneration (due to the accident that killed his parents), and in the position of frontman in the Dalton Academy Warblers (who happen to be the students regarded as the top tier in the social pyramid).
That's how he came to know one Kurt Elijah "Elizabeth" Hummel.