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Choice C is correct. Let $x$ be the 2015 population of Greenville. It's given that the population increased by $7\%$ from 2015 to 2016. The increase in population can be written as $\left(0.07\right)x$. The 2016 population of Greenville is given as the sum of the 2015 population of Greenville and the increase in population from 2015 to 2016. This can be rewritten as $x+\left(0.07\right)x$, or $1.07x$. Therefore, the value of $k$ is $1.07$.

Choice A is incorrect. This is the percent, represented as a decimal, that the population increased from 2015 to 2016, not the value of $k$.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the value of $k$ if the population increased by $70\%$, not $7\%$, from 2015 to 2016.