What can I do to strengthening my knee ligaments?

Strengthen muscles. You can exercise the muscles around your joint to improve stability and strength.
Can't. Ligaments are tough fibrous structures that attach bone to bone. There is not, therefore, a way to actually strengthen your ligaments. You can, however, strengthen the muscles around your knee (quadriceps, hamstrings, and calf muscles) through a dedicated exercise program. Strong muscles will aid the ligaments in stabilizing your knee joint.
Strengthening. You can not strengthen ligaments in and of themselves. In order to provide joint pain relief and stability the best thing is to do resistance exercises. For the knee and lower extremities i suggest tai chi which is gentle on your joint, can be done at home, and is great for strengthening and balance.