Write the signal in its mathematical form (i.e., as a sequence). Your signal may be a long list of numbers. Convert this sequence of numbers to its mathematical expression. For example, if you have the numbers "0, 0.5, 0.25, 0.125 ..." in your signal, recognize this as being produced by the equation (1/2)^k where "k" represents the index of the signal. Thus, your signal's mathematical expression is y(k) = (1/2)^k.
Multiply your signal by z^-k and simplify. The "z" is the variable of the Z-Transform that will be used for analysis after you have constructed the Z-Transform. Just treat it as a variable in the multiplication. For the previous example in which the signal is (1/2)^k, multiply the signal by z^-k, yielding [(1/2)^k][z^-k], which can be further simplified to (2z)^-k.
Add sigma notation. Sigma notation represents the fact that you are dealing with a sequence, not a single number. Write a capital sigma to the right of the term you computed in the previous step. At the top and bottom of the sigma symbol, write the index of the sequence. At the bottom, write 0 -- indicating that the sequence starts at k=0. At the top, write the infinity symbol -- indicating that the sequence goes on without end.
Solve the sigma function. This requires either experience or sophisticated mathematical software, as sigma functions do not have a standard form of computation. For the example in which (2z)^-k is the inner term of the sigma function, an astute observer will notice that this is the geometric sum with the solution [1 -- (2z)^-1]^-1. This solution is the Z-Transform for the signal.