He murdered his daughter in a fit of rage, but was never convicted of the crime. Wanting to undo that tragedy, he worked on creating a time machine so he could send someone back and save her. He needed a psychic to monitor the time traveller, so he spiked some water with a prototype cognition enhancer to jumpstart those abilities in people.
The drug succeeded with Josh Randall, but made him psychotic as a side effect. When the time machine was activated, it went out of control; according to the Fourth Doctor, Finer had only built half a time machine at best, having developed a machine that collapsed probabilities down to a single timeline but lacked a control system to stop itself breaking down everything. Finer tried to turn it off, but Randall used his powers to kill him.
Fortunately, the Doctor and the TARDIS were able to control the machine by having the TARDIS essentially replace the missing components of the machine. When Randall triggered the final control on the machine, the TARDIS was able to contain the destruction to just the specific timeline where these events took place, creating a new timeline where Finer confessed to his daughter's murder immediately after the event. He never tried to build a time machine, and never drugged anyone to generate psychic powers. (PROSE: Psi-ence Fiction)