She was conceived in the TARDIS while in flight through the spacetime vortex (as revealed in a conversation between Vastra & the Doctor in "A Good Man Goes to War"). The Doctor described this as a "primitive" way of creating Time Lords. The Time Lords themselves became the way they were by being exposed to the vortex via the Untempered Schism (referred to in "The Sound of Drums") for many generations.
Because the circumstances of her conception had so much effect on her, she's sometimes called "child of the TARDIS", as if the TARDIS were (figuratively speaking) a third parent.
Technically, she isn't really part Time Lord. She's human with some Time Lord characteristics. She acquired these characteristics in the same way as the Time Lords originally acquired them, not by inheritance from a Time Lord or anyone else.
"A Good Man..." also revealed that Madame Kovarian & her accomplices, working for the Silence, had kidnapped Amy early in her pregnancy in order to be able to manipulate the child's genetics while she was still in the womb & to enhance her Time Lord characteristics. Their objective was to turn the child into a "weapon" against the Doctor.