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I am investigating hair bleach, and want to figure out the redox reaction between $\ce{H2O2}$ (hydrogen peroxide) and $\ce{C18H10N2O4}$ (melanin). $\ce{H2O2}$ is a strong oxidizing agent in this redox equation, but I don't know how to write that out. Someone please help?

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Your formula for melanin is incorrect. Melanin does not have a clear and concise chemical formula; it is a class of natural polymers that differ in their exact composition. The only thing they have in common is that they derive from the amino acid tyrosine and that they at one point polymerised.

Therefore, unfortunately there cannot be an exact description of what happened between melanin and hydrogen peroxide. You can draw a half-equation for the hydrogen peroxide part which would be:

$$\ce{H2O2 + 2 e- + 2 H+ -> 2 H2O}$$

But what exactly happens in melanin and which groups are oxidised is hard to say. What probably happens is a series of radical reactions that in some way oxidises away the aromatic system and thereby moves the absorpion wavelength out of the visible spectrum. But that’s probably about as much as you can say.

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