People have hit on the main two methods you could use--reverse osmosis and distillation. Both methods are energy intensive and don't produce that much water quickly.

Depending on your storage space and how long you want to be prepared for, then IMHO simply storing more water probably makes the most sense rather than trying to desalinate seawater. Sixty gallons is an admirable amount of stored water. Far more than most people. If you can swing that much, then perhaps another 60 is doable for you.