You have to start by producing a set of vectors that span the subspace. The nullspace of your matrix is the solution space of "Ax=0", so the problem boils down to solving this linear system.

Then you must apply the Gram-Schmit process that you mentioned to the vectors you found. If there are more than one vector, you must first construct the orthogonal basis vectors. If there is only one vector you just go ahead and normalize it to obtain the orthonormal basis vector. For example if you have , then

Try it.