Kaikeyi was not evil, she had grey shades in her character .
She is influenced by Manthara because the latter brought out her insecurites saying, if Ram became the crown prince and a king in future , the power would wrest with Kaushalya and Kaikeyi used to treat Kau with a lot of contempt because of this even king Dashratha treat Kau very badly , so what if Kau striked back and made Kaikeyi as her (Kaushalya's slave)?
And after Ram's rule his sons would takeover the kingdom so there was nothing that Bharat would get and he and his children would always remain as princes and had to depend on the king for remuneration.
Later Kaikeyi confesses to Bharata that Ram had done nothing wrong, he could never do anything wrong but she did all this to make Bharat the king