Often, she'd be depicted with enhanced physical and magical abilities. Since she's an assassin, she'd need to be extremely quick and precise. Her knowledge of potions might be used to create poisons or antidotes for her missions. And her ability to apparate would be crucial for getting in and out of dangerous situations quickly.
Yes, there are likely to be such fanfictions. The 'Harry Potter' universe has a vast and creative fanfiction community. Some fans might come up with the idea of Harry Potter being a Muggle assassin as a way to explore different storylines and character developments outside of the original canon. It could involve Harry using his magical skills in a Muggle - centered, assassin - like role, perhaps to protect Muggles from dark magic threats in a more covert way.
One common plot could be revenge. Maybe she's out to avenge the death of someone she loved, using her assassin skills. Another could be that she's part of a secret organization that uses her unique position as Harry Potter to achieve their assassin - related goals.
Perhaps an international wizard who has been trained in both magic and martial arts in some far - flung magical school. This wizard - assassin has his own code of ethics and when he meets Harry, he decides to take him under his wing, believing that Harry can be a force for good in a different way than the traditional Hogwarts - trained wizards.
No. Harry Potter is mainly known for its own magical world and themes of friendship, good vs. evil, etc. Turning Harry Potter into an assassin fanfiction goes against the original spirit of the series and is likely not popular among the mainstream Harry Potter fans.
It could be a rogue Death Eater who has mastered a new form of invisibility spell. This Death Eater might be seeking revenge on those who opposed Voldemort.
Perhaps in a fanfiction, Voldemort has made an alliance with a powerful villain from the Avengers' world and an ancient order of Assassins gone bad. Harry Potter, along with some of his Hogwarts friends, has to convince the Avengers to help them. They would then combine their different abilities. For example, Hermione could use her vast knowledge to figure out the weaknesses of their enemies, while Captain America could lead the group in battle strategy, and the Assassin could provide crucial intel on the secret workings of the enemy alliance.
One possible plotline could be Harry being secretly trained by an unknown figure to become an assassin against the dark forces. He would have to hide this new identity from his friends at Hogwarts.
It depends on your taste. If you like the idea of Harry Potter in a completely different and edgier role like an assassin, it could be interesting. However, some purist fans might not like this deviation from the original concept of the series.
One idea could be a story where a Sith Assassin somehow gets transported to the Harry Potter universe. Maybe they initially see it as a strange and weak world compared to their own, but then they discover the unique magic there. They could try to use their Sith powers to dominate, but find that the magic of Hogwarts and its defenders are not so easy to overcome.