Determine volume of methane used to heat x water to x degrees?
A great place to start would be the heat capacity $C$ of water. It is defined to be the "amount" of heat change divided by the resulting temperature change. In symbols,

$$C=\frac{q}{\Delta T}.$$

Heat capacity is an extensive property; the size of $C$ depends on how much matter there is. We take this into account by dividing the equation by mass $m$.

$$C_s=\frac{q}{m\Delta T}$$

$C_s$ is called specific heat (capacity). Taking $\rho_{\ce{(H2O)}}\approx1$, 

$$C_s \approx \frac{q}{V\Delta T}.$$

Can you take it from here? (Be careful with units.)



Note: It is probably assumed that


the combustion of $\ce{CH4}$ is complete, yielding only $\ce{CO2}$ and $\ce{H2O}$;
density of water does not change over the temperature range;
heat capacity is constant as $\Delta T$ is rather small;
vaporisation is neglible;
no heat lost to surroundings;
decreasing solubility of $\ce{CO2}$ does not contribute to the overall amount of gaseous carbon dioxide (joking...)