Smallest possible integer not obtained from sumset
One possible way is computing the product
$$ \prod_{a \in C} (1+x^a). $$
This is a polynomial $p(x)$.  The coefficient of $x^j$ will be non-zero if and only if $j$ can be represented as the sum of some subset of $C$.  Therefore, given $p(x)$, you can read off the set of all numbers that are obtainable as the sum of some subset, and then find the smallest that isn't obtainable (i.e., the smallest integer $j$ such that $x^j$ has coefficient $0$ in $p(x)$).

As a bonus, you'll also get the number of possible representations of each number which is representable.

The running time is polynomial in $N$, and doubtless you can optimize the degree of the polynomial. (This is dynamic programming in disguise.)

If you're after an algorithm polynomial in the input length (the input can be either $A$ or $C$), that's a different question.  The problem is that the input can be very small; though perhaps such inputs can be handled separately.