Why exactly did Katherine come back to Mystic Falls?

Katherine was only pretending that she came back for Stefan to draw attention away from what she was really doing.
As to her true intentions, the tomb had been opened and the tomb ভ্যাম্পায়ার had escaped. The tomb ভ্যাম্পায়ার all knew that Katherine was in the tomb and as such didn't die when the ভ্যাম্পায়ার in Mystic Falls were rounded up in 1864. The originals, including Klaus, thought that Katherine had died in 1864 and as such Klaus was no longer having people look for her. If any of the tomb ভ্যাম্পায়ার had gotten away and told one of the originals about the fact that Katherine wasn't dead then Klaus would've come after her again. This is why Katherine had John and Isobel destroy all of the tomb vampires. After the tomb ভ্যাম্পায়ার had been destroyed Katherine came back into town to prepare all of the ingredients required for the sacrifice (doppelganger, werewolf, vampire, moonstone, etc.) to try and trade them for her freedom because she thought chances were that word had already gotten out about her not having died to the originals.

Im not certain but i'm pretty sure the Stefan thing was a complete cover up. i think once she saw Elena she all of a sudden became jealous and started using Stefan but the truth is she's always been after the moonstone and the doppelganger (aka. Elena)!

She came back to hand over Elena and the moonstone to Klaus, in the attempt to get back her freedom (I can't remember well but I think it's because she knew about E.'s existance from Isobel)... In the meantime, she has played as usual with everyone, though she says she "loves" Stefan, but this is not the reason why she came back :|