In Ancient Greek, kappa is always a hard sound and never an s sound.
But from an Ancient Greek point of view, English hard k can represent two different sounds: κ (kappa) and χ (khi), i.e. in different contexts English k can either be aspirated or unaspirated, pronounced with or without an 'h' sound. K in English is typically aspirated, when in the beginning of a word or a stressed syllable, and otherwise unaspirated. English speaking people don't necessarily even notice a difference between these two sounds, but for an Ancient Greek they were two different sounds and represented by two different letters.
To an Ancient Greek, these English would have been written with a khi, not a kappa: kill, cool, chaos, cap, cop, Ken.
But an Ancient Greek would have written these English k sounds with a kappa: skip, school, scar, taken, Christ.
So if you (I assume you're a native English speaker) don't pay attention when pronouncing Κιλικία, you actually pronounce it Χιλικία!
It the same way, Ancient Greeks made a difference between aspirated and unaspirated p and t sounds.
π: spit, spoon, spade, cap, pride
φ: pit, put, pen
τ: stop, get, trip
θ: top, taken, tap
Imagine a French guy speaking English - that's how you should pronounce pi, kappa and tau!