How to sample from a given univariate CDF is a huge subject, so I will assume that part of the answer is known and will address how to find the conditional CDF from the copula.
By definition, any copula assigns probabilities to rectangular regions (within the unit square) delimited on the right by its first argument and above by its second argument. In particular, when $U$ and $V$ are uniformly distributed with $C$ as the copula for $(U,V)$ and $0 \lt \epsilon \le 1 - u$ is sufficiently small,
$$\eqalign{
\Pr(U\in (u, u+\epsilon]\text{ and }V \le v) &= \Pr(U\le u+\epsilon, V \le v) - \Pr(U\le u, V \le v) \\
&=C(u+\epsilon, v) - C(u, v).
}$$
Therefore, the conditional cumulative distribution function ought to arise as the (right-hand) limiting value of
$$\Pr(U\in (u, u+\epsilon]\text{ and }V \le v\,\Big|\,U\in (u, u+\epsilon]) = \frac{C(u+\epsilon, v) - C(u, v)}{\epsilon}.$$
Provided this limit exists (which it will almost everywhere for $u$), by definition it is the first partial derivative, $\partial C(u,v)/\partial u$. This, therefore, gives the conditional CDF for $V\,\Big|\, U=u$ evaluated at $v$.
The left figure shows a contour plot of the copula (representing a surface) $C(u,v)=uv/(u+v-uv)$. The right figure is the graph of the conditional distribution of $V$ for $u\approx 0.23$. It is a cross section of the rightward slope of the surface.
Reference
Roger B. Nelsen, An Introduction to Copulas, Second Edition. Springer 2006: Section 2.9, Random Variate Generation.
For a copula that corresponds to a known multivariate distribution, you can simulate from that distribution and then make the margins uniform (e.g. Gaussian copula, t-copula).
More generally if you can work out the conditional (either $C(u|v)$ or $c(u|v)$), you can simulate from a uniform for $V$ and then from the conditional, perhaps via inverse-cdf (if you know $C(u|v)$) or perhaps via say accept-reject (maybe an adaptive accept-reject, some version of ziggurat, etc, if you know $c(u|v)$).
In the case of bivariate Archimedean copulas, following Nelsen (1999) or Embrechts et al., (2001), we have a mechanism for then generating from them as follows. Suppose $(U_1,U_2)$ has a two-dimensional Archimedean copula with generator $\phi$. Then:
Simulate two independent $U(0,1)$ random variables, $v_1$ and $v_2$
Set $t=K_C^{-1}(v_2)\,$, where $K_C(t)=t-\phi(t)/\phi'(t)$
The desired simulated values are $u_1=\phi^{-1}(v_1\,\phi(t))$ and
$u_2=\phi^{-1}((1-v_1)\phi(t))$.
There are other methods; for example in some cases it might sometimes be practical to do some version of bivariate accept-reject, say, or via transformation to some convenient bivariate distribution on which accept-reject might be applied.
Best Answer
A typical approach (see e.g. Nelsen 2006, p. 41) is to sample two independent uniform distributed random vectors $u$ and $y$ of the desired sample length. The conditional copula $C_u$ (conditioned on $u$) is given through the partial derivative: $$ C_u(v) = \frac{\partial}{\partial u} C(u,v) $$ Hence, one needs to solve $C_u(v)=y$ for $v$ to get the desired pair $(u,v)$. For a "custom made" copula, one has to calculate its partial derivative and its quasi-inverse. In case the copula is not completely "custom made" it might already be covered in other statistical software. One might for instance take a look into the R packages copula and VineCopula offering a rich set of families (speaking from my R experience, there are more in R and of course in other languages).