This depends on what do you need it for. You have to answer question such as
- Do you want to describe motion at non-relativistic velocities near the galaxy?
- Does the motion occur inside the galaxy or outside?
- What do you assume about the morphology of the galaxy? (What type of Galaxy is it?)
If the answer to 1. is yes, you can typically use the metric in the Newtonian limit, i.e.
$$ds^2 = -(1 + 2 \Phi) dt^2 + (1 - 2 \Phi)(dx^2 + dy^2 + dz^2)$$
where $\Phi$ is the Newtonian potential sourced by the galactic matter. Now you "just" need to find the right $\Phi$. The answer depends a lot on questions 2. and 3. For instance, an almost spherically symmetric galaxy can be modeled by a spherically symmetric potential that is given simply by $\Phi = -GM(r)/r$, where $M(r)$ is the mass enclosed in the radius $r$. On the other hand, the outside potential of a flattened, almost axially symmetric galaxy can be modeled by potentials such as the Vinti potential or generally a multipolar expansion.