You need to avoid the main holiday times - Christmas, Summer etc.I would think February/March or October.
Ive never noticed it making a difference. The amt of time you book in advance makes a difference. You can play around with the scheduling tools on any airlines. Just dont go as far as the credit card payment page or put in your name.On the airlines website, be sure to click Sort by lowest fare first They dont readily let you know you can sort like this.( On American Airlines, you have to click sort by Price and Schedule. THEN, sort by lowest price first.) It can sometimes make a price diff of $300. ( also know the price is just one way ..then you go to your return flight purchase)