Ah, Goðafoss: waterfall of the gods. Its historical significance (this is the spot where, very famously, Iceland's religious allegiance was decided in the year 1000), coupled with its easy accessibility (located directly off the Ring Road), have made this 12 m tall, 30 m wide deluge one of the most sought-after natural attractions in the country. In an age of Icelandic cruse ships and tour buses, however, this is not necessarily a good thing. But all hope is not lost if you're looking to spend a little alone time with this beautiful and noteworthy waterfall. For starters, plan to visit either early in the morning or later in the evening when most tourists are favoring the warmth of the indoors and you'll be off to a good, isolated start. Then, instead of settling for the view from the west bank, hike down stream to the foot bridge and cross over to the east bank of the falls. Since it requires a longer hike and offers a much smaller parking lot, this side tends to be far more deserted, even though the views are arguably better.
If you're looking for more of a visceral experience, it might interest you to know that Goðafoss's section of the Skjálfandafljót River has become increasingly popular with whitewater kayakers. And yes, this includes the falls itself. Should plummeting 40 ft into the frigid waters below be more your adventurous style, stop into nearby Akureyri and seek out some advice from knowledgeable adventurers there. Then all we can say is "good luck."