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.