Kikyou
…
Lies
Kikyou knows perfectly well that clumsy Kagome is not her reincarnation; that people think the silly child is her reborn is a result of physical resemblance and wishful thinking on Kaede and Inuyasha's part.
She only calls Kagome 'my reincarnation' to irritate the girl and prompt Inuyasha to be more attentive to her rather than run after the scandalously dressed interloper.
At least she dresses modestly now and knows which way up to hold a bow.
Hatred
Kikyou hates Kagome because it is a shard of the untrained miko's soul that binds her to her clay body, forcing her to devour the souls of other maidens or fade away to nothing in her pottery shell.
Kagome's soul fragment is like a rope wrapped tightly around her body and limbs, strangling Kikyou's abused soul as it binds it in place.
Agony
Kikyou can sense the shikon shards because the chunk of Kagome keeping her from her rest resonates with both the jewel and the air-headed would-be miko herself. Which hurts terribly whenever she approaches either a shard or Kagome herself, although the latter effect has been significantly dampened of late.
Nonetheless, this means that Kikyou is reminded of her tormentor wherever she goes, as the shards are scattered far and wide. She only helps the Inutachi find them so that Kagome can repair the jewel and go away.
Oh, and for Inuyasha's sake.
Deduction
Kikyou is nobody's fool; the constant pain dulls her mind but she is accustomed to it now and it only slows her down a little. Kagome's unnatural affinity for the Shikon gives the undead miko an unpleasant inkling of whose reincarnation the childish interloper really is.
That the foolish, naïve, weak girl may actually be Midoriko reborn makes her detest the silly, hapless child even more.
Kikyou is obsevant, too. She also doesn't like Kagome very much, but she's the passive-agressive type except when the enemy is a demon. Then she gets to kill it.