Natasha was no longer under her care and was placed under the care of another person to teach her because June was probably going to murder her, normally they wouldn't mind if she was an ordinary soldier, but she has one of the brightest brains.
June was busy studying the DNA of the soldiers who exploded from Natasha's serum that night and saw the unstable DNA eating up the normal human DNA and turning it into that.
She received memories from her future self, which was incredibly painful, meaning it was years worth of memories. It came from five years into the future and a young boy that looked like Columbus... That is if she had known him years before they met and if he had grey eyes.
That was the color of her eyes. So this means that she would use her DNA with Columbus' DNA, but how did she manage to stabilize it with the chaotic DNA of the beast?