R offers a package called copula (http://cran.r-project.org/web/packages/copula/index.html). This package allows you to handle copula in an very easy way.
First of all, I assume you have a covariance matrix called covSample and CovBetter (for your new one. In the example, I will assume they have 8 dimensions and you want 5 degrees of freedom for the tCopula. You can create copulas using them by:
corSample <- cov2cor(covSample)
corBetter <- cov2cor(covBetter)
copG <- normalCopula(param=corr[lower.tri(corSample)],dim=8,dispstr='un')
copT <- tCopula(param=corr[lower.tri(corBetter)],dim=8,dispstr='un',df=5)
uG <- rCopula(10000,copG)
uT <- rCopula(10000,copT)
After that, you should have a uG and a uT matrix that you can put into your inverse margin distribution and analyse whatever you are interested in.