Alright, this takes place between the fourth season finale and the ending two part special where David Tenant is regenerated. I hope I just didn't spoil the end for you!
DICLAIMER- I don't own anything apart from my original character
In the season finale…
Captain Jack Harkness ran to catch up with Mickey and Martha. On an afterthought, he ran back to the Doctor.
"Doctor," he said. "I need you to do me a favor."
"Sure. Well… It depends…" he answered.
"I have a daughter, she should be about twenty now. She was born when I was still waiting for you. I haven't seen her since she was one. If you ever find a girl named Grace Feldsparr, just, take her with you. Trust me; she'll be able to handle it without any explanation."
"Okay… Yeah, I'll do that."
He ran back to the two who had stopped to wait for him.
"Hm," The Doctor thought out loud.
Several month later…
The Doctor was back in London. He didn't have anyone he wanted to talk to. He just wanted to walk around. He loved London, with all its memorable people and places. He was heading back to the TARDIS when he heard a commotion. Running quickly in his own peculiar way, he reached the disturbance. Some kid had grabbed a woman's bag. He was about to intercept when the woman grabbed the kid's arm and yanked back the bag.
"You don't want to do that kid. How old are you?" she said sternly.
"F-Fifteen ma'am." He answered timidly, unsure what was going to happen next.
"Why did you steal my bag?"
"I'm hungry ma'am."
"All you had to do was ask! Now, go down the street where there's a restaurant, real nice place. Go inside and find a guy named Tim, tell him Grace sent you. You'll get a meal and a job if you need the money."
"Th-Thank you ma'am!"
"It was no problem; just don't let me ever catch you stealing again!"
The kid ran off down the street. The Doctor was staring silently. The woman turned around and caught his gaze.
"Something wrong sir?" she asked gently.
"No, no, nothing's wrong. Just wondering, what's your name?"
"Grace Feldsparr. You are…?"
"Feldsparr, Feldsparr… Is your father Jack Harkness?"
"How do you know my dad?" her eyes widened. "You're the Doctor!"
"That's me. It's nice to be recognized once in a while."
"It was real? Oh gosh, I need to sit down, although I probably shouldn't do that in the middle of the sidewalk. Would you like a cup of tea? My place isn't far from here."
Grace led him one block until they reached a crumbling building. Next to it was a single concrete wall. A couple of kids were using spray-paint to create graffiti.
"Hey! Trent! Alex! What do you think you're doing?" Grace yelled. "We've talked about this you two. Now, you know where I keep the paint, go repaint that wall now!"
"They walked off grumbling something."
"Wow." The Doctor said.
"Even though I'm only twenty, I'm the authority around here. It's actually pretty cool."
She forcefully put the key in the lock and turned forcefully. "Crummy lock, it always gets stuck." After three flights of stairs, there was another door. She opened that one with a different key and led him inside.
"I know its small, but I can't afford much. College, debt, food, and insurance cost a lot. I'll go put the kettle on, make yourself comfortable." she left to the tiny kitchenette off of the room. The Doctor studied a few photos on the mantelpiece. One was two girls and a guy smiling at a café, once of them was Grace. The other photos showed a middle aged woman with brunette hair and glasses. Grace reentered and handed him a mug.
"So what did you mean by 'it's all real'?" he asked.
"My father left me a package of letters addressed to me. They were full of stories about his life, Doctor, and you were a major part of it. Your TARDIS, the feral planet, the last humans, Martha, Rose, WWII, everything."
"That's pretty strange."
"You tell me. Most kids here grew up with the Three Blind Mice and Cinderella. I grew up with the Daleks and other aliens."
"Sounds like my childhood, almost, but not exactly though. Anyway, what are you studying?"
"I'm majoring in Archaeology and getting a minor in Egyptology. I've only been in college a couple of years, but I still have a very long way to go."
"Interesting… I want to show you something."
The Doctor went back down the stairs hurriedly, across the street, and back to the TARDIS. Grace kept up easily and stopped in her tracks when she saw the blue phone box. He opened the door and gestured her inside. She walked in, mouth agape.
"It's beautiful." Only thing she said.
"Yes she is." The Doctor replied. "Yes she is."
I've always wanted to make a Doctor Who fan fiction, but I haven't been able to come up with a good character and storyline until now.