For a curve over an algebraically closed field that is complete with respect to a nontrivial valuation, we study its tropical Jacobian. We first tropicalize the curve and then use the weighted metric graph to compute the tropical Jacobian. Finding the abstract tropicalization of a general curve defined by polynomial equations is difficult, because an embedded tropicalization may not be faithful, and there is no known algorithm for carrying out semistable reduction. We solve these problems for hyperelliptic curves by using admissible covers. We also calculate the period matrix from a weighted metric graph, which gives the tropical Jacobian and tropical theta divisor. Lastly, we look at how to compute a curve that has a given period matrix.