For example example of a pdf or pmf with correlation . It is obviously easy to find ones with 0,1,-1. But what would be a good general way to compute one with a specific value?

You're asking a very general question. If your goal is just to generate correlated observations with a set correlation you could look into the bivariate normal distribution.

It seems that you are having a set of independent random variables and want a copula to give them a specific correlation structure.

Dason has pointed out a common example - the multivariate normal copula which uses the covariance matrix as an input of parameter. In general this will not give you the answer directly as it involve some nonlinear transformation (probability integral transform)

Another very simple example is that you may consider the Cholesky decomposition of your desired correlation matrix. Then by applying the resulting linear transformation you can always achieve the desired correlation structure (with proper scaling of course). The limitation is that it is a linear transformation of the random variable - anyway it should be a trivial example to start with.