If $(X,Y)$ is a random vector, then the mean of the random vector is defined as $(E[X], E[Y])$, so you just need to take the mean of each component separately. If $(X,Y)$ has a joint density $f$, then for instance $$E[X] = \int_{-\infty}^\infty\int_{-\infty}^\infty x f(x,y) \, dx \, dy.$$ Note that in cases where $X$ and $Y$ are independent, the joint density decomposes as $f(x,y) = g(x) h(y)$ into the marginal densities, so the above integral would just be the usual formula $E[X] = \int_{-\infty}^\infty x g(x) \, dx$.