She intends to take over the current royal family from within, and then elevate herself to become the most powerful person in the country by proxy of her son, Arlendo, becoming the new emperor. I believe she partly sees it as revenge because her own kingdom was conquered by the empire back then. (You took over my country, so I'll take over it from within.)
The empress wants to get rid of the duchess because not only are their beliefs polar opposites, it's also because the duchess is fiercely loyal to the current emperor. (Emperor and duchess have been best friends since their youth.)