You've probably realised already that a possible side reaction is the SN2 nucleophilic substitution of Cl by HO-, which would look like this:
Evidently, this is not the product we actually get. Hydroxide ions are considered 'hard' nucleophiles, as they have a high negative charge density, and they tend to react best with 'hard' electrophiles with high positive charge density. The electrophilic carbon here is reasonably 'soft' whilst the proton adjacent to it is considered 'hard' due to its small size (you can learn more about this here). As a result, hydroxide ions like to act as bases instead of nucleophiles, and the mechanism favoured is not the SN2 shown above, but in fact the competing E2 elimination shown below.
This proton may not look particularly acidic, but formation of a new alkene alongside loss of a reasonably stable chloride leaving group drives the deprotonation.
Another contributing factor is that whilst the substitution has a reasonably small change in entropy, the elimination results in a greater number of species than in the reactants, so has a large positive entropy. This is relevant because Wolff-Kishner reduction is usually done at a reasonably high temperature, and entropy becomes a greater influence on the spontaneity of a mechanism as the temperature increases. This further favours E2 over SN2.
The chloride is considered a base-sensitive group for the above reasons, and this second pathway is inevitable alongside a Wolff-Kishner reduction in a highly basic solution.