From Doctor Who Magazine issue #504, in the section that allows fans to ask the then showrunner Steven Moffat their questions, he has this to say:
FINLAY WORRALLO asks: Why did the TARDIS dislike Clara?
The TARDIS, being aware of all time simultaneously, was also aware that Clara was the precise motivator that would drive the Doctor to an extreme that was dangerous for all time and space. She knew, that although the Doctor loved her very much, she was bad for him and that the coming of the Hybrid would be the result of their association.
From this, it could be inferred that the association of the Doctor and Clara together form the Hybrid. Time Lords and Humans are both referred to as warrior races in the episode (see the ending dialogue between Me and the Doctor), which would fit the Hybrid's description. The billions of hearts they would break would presumably be the billions that would die if
Clara was kept from dying, her death being a fixed point, and preventing fixed points tears the universe apart (see The Wedding of River Song).