There are several approaches.
Use linear algebra. Compute Schur polynomials (using the Jacobi-Trudi identity, say), and then use the fact that the coefficient of $s_\nu$ in the product $s_\lambda s_\mu$ is a Littlewood-Richardson coefficient.
Use a combinatorial interpretation.
One can count so called Littlewood-Richardson tableaux, or lattice points in so-called Berenstein-Zelevinsky polytopes. The latter can be done rather efficient using some lattice-point counting program, such as lattE.
However, note that it has been proved that computing Littewood-Richardson coefficients is #$P$-complete, meaning that there is no super-efficient algorithm for computing these (that is, no nice closed formula).
If you want to compute several coefficients, a recursive approach might be suitable. One can show that the multiplicative constants for so called shifted Schur functions are generalizations of Littlewood-Richardson coefficients, and these do satisfy a very nice recursive formula.
See the paper and proposition 3.4 in
A Littlewood-Richardson Rule for Factorial Schur Functions
by Alexander I. Molev and Bruce E. Sagan.
See also this answer.