Solution:

Write the product as $\prod_{k=1}^{15} (1 + a_k x)$ with $a_k = (-1)^k k.$ An $x^2$ term arises by choosing the $x$-term from two factors, so $C = \sum_{i \lt j} a_i a_j,$ and $$C = \frac{\left(\sum a_k\right)^2 - \sum a_k^2}{2}.$$

The alternating sum is $(-1 + 2) + (-3 + 4) + \cdots + (-13 + 14) - 15 = 7 - 15 = -8,$ and $\sum a_k^2 = 1^2 + 2^2 + \cdots + 15^2 = \frac{15 \cdot 16 \cdot 31}{6} = 1240.$

Thus $C = \frac{64 - 1240}{2} = -588,$ so $|C| = 588.$