Driving on the ring road, look out for a sign saying '8 Heinabergsjökull ⌘' (GPS coords: 64.249970, -15.647018). From there it's about 8km to the glacier lagoon. We tried driving there with a regular car, however we parked the car halfway due to the gravel getting worse and worse. Luckily for us we were able to hitch a ride with a very nice American couple who had the same car as we did, but the amount of rocks that got slammed to the car's underside really made me confident that leaving our car behind was the right choice.
Once you get to the glacier lagoon it's very likely that you'll have it completely to yourself as it is far from a popular spot, but it's very pretty when we were there!