The Old Latin alphabet had 3 letters for the sound [k]: C, K, and Q. K was used before A, Q before V (the shape U appeared later), and C elsewhere. Besides, C was also used for the sound [g]. Later, K was marginalized and used only in a couple of words, e.g. Kalendae, and a new letter, G, derived from C, was introduced for the sound [g].
In the post-classical Latin dialects, the sound [k] before [i] (spelled I and Y) and [e/ɛ] (spelled E, AE, OE) changed into [s], or [ts], or [tʃ] all of which can be found in the modern Romance languages.
Since the European languages which now use the Latin script are or were once Roman Catholic, and the speakers were aware of the strong association of Q with U, and of the fact that C before E, I, and Y is not pronounced [k], and that the only universal letter that means [k] in all the positions was K, it was this K that they chose for the use in their native languages to denote the sound [k].