Understand the roles of the units in factoring Gaussian primes. For example, 5 is composite in the system of Gaussian integers because 5 = (1 + 2i)(1 - 2i), but it is also true that 5 = (2 + i)(2 - i). This is not really two different factorizations because the factors 1 + 2i and 1 - 2i can be transformed into 2 + i and 2 - i by multiplications by units and so are considered the same for factoring purposes. For example, (i)(1 + 2i) = i - 2 and (-1)(i - 2) = 2 - i.
Find the norm of the Gaussian integer you want to factor. The norm of a complex number a + bi is a^2 + b^2. If A = B X C then Norm(A) = Norm(B) X Norm(C). This narrows down the list of possible candidates. For example, the norm of 5 is 5^2 + 0^2 = 25 and the only factor of 25 is 5. This means we only need to look at candidates a + bi such that a^2 + b^2 = 5. The obvious candidates are 1 + 2i and 1 - 2i, both of which have norms 1^2 + 2^2 = 1 + 4 = 5.
Try each of the candidates to see if any of them divide the number you are factoring without leaving a remainder. The quotient may produce another Gaussian integer that may or may not be prime. Notice that if a + bi is a factor, you do not have to consider b + ai or b - ai as these are just one or more of the units multiplied by a + bi. When all of your factors are prime, you have the unique factorization.