To add to Gerald Edgar’s answer:
1) Kurzweil–Henstock integral satisfies $\int_a^bf(x)\\,dx=F(b)-F(a)$ even under the weaker assumptions that $F$ is continuous and $F'(x)=f(x)$ for all but countably many $x\in[a,b]$, hence it fully subsumes the integral you want to define. It is, however, strictly more general: if $A\subseteq[a,b]$ is a null set and $f$ its characteristic function, then $\int_a^bf(x)=0$ (as a Kurzweil–Henstock or Lebesgue integral), so the only choice would be $F$ constant, but then $0=F'(x)=f(x)$ only holds outside $A$. Thus, in general, the set of exceptions may be an arbitrary null set. (It cannot be any worse: the Lebesgue differentiation theorem mentioned by Gerald Edgar holds for Kurzweil–Henstock integral as well.) In particular, your integral (unlike Kurzweil–Henstock integral) does not extend Lebesgue integral. It does not even extend Riemann integral, as we can take for $A$ a closed set.
3) If we allow $C$ to be an arbitrary null set, then the definition no longer makes sense: if we take for $F$ the Cantor function, then $F$ is continuous and $F'(x)=0$ almost everywhere, so we would be forced to put $0=\int_0^10\\,dx=F(1)-F(0)=1$.
One can salvage the definition by making stronger requirements on the function $F$. That is, we can put $\int_a^bf(x)\\,dx=F(b)-F(a)$ if $F'(x)=f(x)$ for all $x\in[a,b]$ except a null set, and $F$ is an admissible indefinite integral function. One can obtain a characterization of Lebesgue integral in this way by taking $F$ absolutely continuous. For Kurzweil–Henstock integral, one needs a broader class of functions, whose description is a bit more delicate, see e.g. http://link.springer.com/article/10.1007%2Fs10587-008-0081-0 .