The thing that appears at the end of the Genocide route being the manifestation of your bloodlust, or in more general terms, your need for gratification via experience points, leveling up, and discovering new content (dopamine release, really), which at that point would have grown so strong, you could no longer control it.

You're neither.

You're definitely not Frisk as their name is always constant. They are a kind child that tries to show mercy to everyone, and in Pacifist, you enable them to do so.

Chara is a character from Undertale that you name. Chara already exists in the story even if you never started the game.
In Pacifist and Neutral, they reside in Frisk and narrate Frisk's adventure throughout the Underground while you control Frisk.
In Genocide, powered by the LOVE you gain, Chara learns to directly control Frisk at times, at points where they don't emote to Sans's whoopie-in-the-hand, walk through Papyrus's electricity maze, and kill Sans and Asgore. (Fun fact, you're still the one who kills Flowey. If you don't press Z, Chara never attacks him as he begs for mercy.)

So, what are you? You are "The Player", or "The Anomaly", an all-powerful third entity outside of the world of Undertale, separate from Frisk and Chara.
Your SOUL and Determination is what allows Frisk to SAVE and LOAD, and after True Pacifist, you are the one who can use True Reset to erase everything you and Frisk have worked towards. Your actions affect Frisk's LOVE and Chara's power, and in Genocide, this comes to bite you in the ass.