A k-space is a compactly generated Hausdorff topological space. (I used the terminology "k-space" in the question, in order keep the question within the limit of 150 characters.)
Note that under the stated hypotheses in the question, $X$ is automatically Hausdorff and $f$ is automatically a closed map.
Thus, in more basic terms, the question is: If $X$ and $Y$ are topological spaces, with $Y$ being compactly generated and Hausdorff, and $f : X \to Y$ is a proper covering map, then is $X$ necessarily compactly generated?
If the answer is negative, is there a positive answer if $X$ is also assumed to be connected and locally path-connected (both being common assumptions on covering spaces)?
A particular case, in which the answer is positive, is when $Y$ is (Hausdorff and) locally compact. In this case, even fewer hypotheses suffice; for it is well known that if $X$ and $Y$ are Hausdorff, with $Y$ being locally compact, and $f : X \to Y$ is a continuous, proper, surjection, then $X$ is also locally compact. (The converse is also true). Does this result also hold if we replace the hypothesis, "locally compact", with "compactly generated"?