It has been mentioned that the first Death Star had a class 4 Hyperdrive and a class 20 backup, so it could move from system to system (so it could destroy targets.) As the weapons system of the First Order's Starkiller base is partly based off of the Death Star weapons system, does Starkiller base have a hyperdrive to move?
No, it was made stationary
Starkiller was built stationary on a planet. Also, it used hyperspace to target systems very far away, thus eliminating the need to move in order to fire. It is never mentioned that the station was moved from one location to another, but secretly built on a remote planet in unknown areas of the Galaxy.
The planet-station is mentioned as 'mobile', but only rocket ports are used for that. It is unlikely that such technology could be effectively used to move the planet across the galaxy, but rather for orbit correction and rotation (as in real life space stations), so that the firing side can be properly adjusted.
From Star Wars: Force Awakening official novelization eBook by Alan Dean Foster, Chapter X (emphasis mine)
The fleet of Star Destroyers stood off the white world. Spectacular and isolated, with a mean surface temperature varying from merely cold to permanently arctic, the planet had been altered: its mountains tunneled into, its glaciers hacked, and its valleys modified until it no longer resembled its original naturally eroded form. Those who had remade it had renamed it.
Chapter XIV Finn:
“General Hux told us it’s the most powerful weapon ever built. He said that it can reach halfway across the galaxy.” Fresh murmurs of disbelief greeted this latest assertion. “And in real time. Because it doesn’t reach across the galaxy; it reaches through it.”
From Wookipedia article on Starkiller base: (emphasis mine)
Starkiller Base was a mobile ice planet located in the Unknown Regions that served as a stronghold of the First Order thirty years after the Battle of Endor. Commanded by an unofficial triumvirate of General Armitage Hux, Kylo Ren and Captain Phasma, it was armed with a superweapon capable of destroying entire star systems halfway across the galaxy. The First Order viewed this ability as an evolution of the planet-destroying capabilities of the Galactic Empire's Death Star designs, with the weapon serving as the ultimate culmination of Old Empire research into dark energy translations and hyperspace tunneling.
10 Facts on The Force Awakens - Empire Magazine, quoted in the article above
Fun facts that aren't revealed in the film: 1) the First Order chose this planet because of its minerals, and 2) yes, it can move, since it has rocket ports on its far side