Experimentally finding order of a reaction
There are a couple of odd things here. 


the time for a reaction to be complete is usually infinite in theoretical kinetics, so I don't see how it could be useful and/or relevant. One usually looks at the initial rate of a reaction.
if bleach is in large excess, you may have a degenerate order. That means that the concentration of bleach barely varies during the reaction and can be considered a constant.
you use an expression with an infinitesimal amount of time ($\mathrm{d}t$), when $\mathrm{d}t$ is everything but infinitely small... 


Let me note bleach $\mathrm{ClO^-}$, and bromocresol green "BCG".

I assume that the reaction is monitored through the changes in absorbance, that is, the concentration of BCG. Its rate $r$ is the rate at which BCG is disappearing.

$$r = - \frac{\mathrm{d}[\mathrm{BCG}]}{\mathrm{d} t} = k \times [\mathrm{ClO^-}]^\alpha \times  [\mathrm{BCG}]^\beta $$ where $\alpha$ and $\beta$ are the partial orders.
If you use two different concentrations of bleach ($C_1$ and $C_2$), the ratio of rates will be :
$$\frac{r_1}{r_2} = \frac{k \times (C_1)^\alpha \times  [\mathrm{BCG}]^\beta}{k \times (C_2)^\alpha \times  [\mathrm{BCG}]^\beta} = \left(\frac{C_1}{C_2}\right)^\alpha$$

A simple way to get the initial rate is to measure the slope of the tangent to the graph of  ${[\mathrm{BCG}]}$ along time (dotted line).



This line crosses the time axis at $t=\Delta t$.

In this case, 
$$ r_i(t=0) = \frac{\mathrm{[BCG] (t=0)}}{\Delta t_i}$$
Therefore, one can write  :
$$\frac{r_1}{r_2} = \left(\frac{C_1}{C_2}\right)^\alpha =  \frac{\Delta t_2}{\Delta t_1}$$
which kind of looks like the formula you were given.