The statement that this always CAN be done is called 'Ado's theorem'. One way to get an answer is to look for proofs of this theorem online. This blog post by Terrence Tao looks useful (but I didn't read it all): https://terrytao.wordpress.com/2011/05/10/ados-theorem/.

EDITTED IN: A good start (see also the blogpost) is to first try and find the center of the Lie algebra, so all linear combinations of your operators that have the property of commuting with all the operators. If your center is only the zero operator, things are very simple. And if it is one dimensional things are not that bad either.