This documentary made shortly after WW2 goes into detail over how V2 operations worked: https://www.youtube.com/watch?v=80DzifHHIxk&t=7m18s. The guidance system is discussed shortly after the 7 minute mark, including footage of an on ground mockup / test rig.

To summarize: the rocket has two gyroscopes. These detect any rotation the rocket does, one for horizontal, the other for vertical. They're linked to electric actuators that move control flaps. There are 4 flaps in the exhaust nozzle of the rocket, and 4 smaller trip flaps on the rocket fins. The guidance "logic" is a simple circuit that keeps the rocket at a specific pitch of 43 degrees. The rocket was aimed in azimuth by precisely positioning it over survey marks at the launch site, pointing fin 1 directly at the target. Range was controlled by shutting off the engine at a precise velocity, as measured by a 3rd gyro system acting as an integrating accelerometer. The guidance was only active while the rocket was running, for the first 10% or so of the flight. After that the V2 was just a ballistic projectile.

Later versions of the V2 used some more advanced guidance that isn't covered in that film. The V2's sent towards Antwerp had antennas and followed a radio guidance beam. Again, this was just to keep the rocket on a simple specific pitch and heading, not full command guidance like modern systems. Likewise the engine shut off timing could be sent by radio in later versions. They didn't have any form of telemetry sending data back. The US clone of the V2 used as a sounding rocket, however did add this.

Overall the inaccuracy of the V2 limited its usefulness as a strategic weapon. They would often miss their target by 10's of miles. Ultimately they were mostly used as a psychological weapon targeting large dense population centers.

To answer /u/throwaway352432 's question, yes, British intelligence did successfully trick the Germans. They falsely reported that the rockets were overshooting London considerably, and then once the Germans were tricked into changing their aim, reported false mass casualties to keep them convinced.

Adding to this question, what sort of post-launch information were the Germans able to get? Was there any data to help them make adjustments to the next wave?