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I didn't understand how to simulate asset prices by using non normal random numbers.
I am assuming that it would be incorrect to use the standard Geometric Brownian Motion, since it is based solely on normally distributed random variables z~N(0,dt).
As discussed below it should be a good idea to simply assume that the random numbers are logreturns.
For non-normal asset price models you could look at the theory of Lévy-processes.
If we assume that you work in the physical probability measure $P$ and that the random numbers that you have generated are daily log-returns, then you can do the following: Asset $i$ has starting price $S_0^i$ and for the future prices you can put $$ S_t^i = S_0^i \exp(\sum_{k=1}^t r^i_k), $$ where the $(r_k^i)_{k=1}^N$ are the sampled returns from the copula that you have described.
EDIT after comment by the OP:
If you want to price an option then you can sample the paths but with a drift equal to the risk free rate. Thus you subtract the expected value (depending on your distribution) and add the risk free rate. If you look at the theory of Lévy processes this means that you calculate the compensator (for the theory of Lévy processes you can look here.
Usually in the context of non-Gaussian models options are priced using Fourier-transform techniques. To learn the standard theory of these processes you would work through "Financial Modelling with Jump Processes" by Cont and Tankov.
Note that using a different distribution for the log-returns than a Gaussian you already model jumps of the stock price (very small ones).
