Spanier's book is relatively old (so I know it does not quite answer your question), but excellent. It uses category theory from the get-go. Riehl's "Categorical homotopy theory" is very well-written, though it may be a bit too advanced if you hadn't seen a bit of algebraic topology already. Riehl's book is focused on the categorical aspect via Quillen model structures. A major and important area of algebraic topology. The categorical requirements for appreciating the Quillen machinery are not modest, and the book does a great job presenting all that is needed.
Having said all that, if you find a text that you like but which avoids the category theoretic language, then you should be able to quite easily fill-in the gaps on your own. Simply consult online sources (e.g., the nLab) to get the categorical pictures (and then some) of whatever concept you are learning. In an introductory text you will probably cover the fundamental group(oid), Van Kampen's Theorem, some higher homotopy groups, and some homology. It should be easy to find the categorical/functorial description of these concepts no matter how the book presents it.