The easiest (and one of the cheapest) ways to get to Reykjavík from Keflavík International Airport is by bus.

The bus leaves right next to the airport and can take you to the main bus terminal in Reykjavík or straight to your hotel (and vice versa on your way back).

The link in Book now is for Airport-Hotel but below are links for other options.

Hotel to Airport -

Airport to BSÍ

BSÍ to Airport