Winkler's method in the absence of iodide
Bad news is that Winkler's method will fail in the presence of bleach or any other oxidizing agent besides oxygen. Neither sodium thiosulfate nor potassium iodide are compatible with bleach, whether it is a oxygen bleach or a chlorine bleach.


  I used $\ce{NaOH},$ $\ce{MnSO4},$ $\ce{H2SO4}$ in this experiment (and $\ce{Na2S2O3}$ as the titrant) and I'm not too sure whether I used potassium iodide. However, I've obtained a result and found out a relationship where the dissolved oxygen content increases as I add more bleach into the water.


As a good scientist of future, you always write whatever you are doing at the time of experiment in your notes.

Your results are skewed towards higher numbers because sodium thiosulfate is being consumed by released iodine not only from the oxygen reaction but also from the bleach.