During the 2100s, he discovered someone from the 26th century who traveled in a time-pod. In light of his failures to invent anything significant, he saw an opportunity to usurp the craft in order to use it to collect future technology for dismantling and study in hopes of recreating it in his own time and taking credit for the inventions he purloined. He stole the craft from its rightful user and spent the next few weeks understanding how it worked. He then used it to travel forward in time, finding himself in 2368.
In 2368, Rasmussen visited the USS Enterprise-D claiming to be a professor of history from the future. Despite his putting on a cheery and eccentric demeanor, the crew had a hard time believing this initially and tried to avoid asking questions regarding the future. Picard, however, allowed him on the ship and to distribute questionnaires to the crew. Rasmussen hinted at being interested in their current mission in saving a planet to help the guise. Eventually, however, Captain Picard asked for Rasmussen's help in what action he should take. He handled the philosophical conversation well, saying he can't affect Picard's decision.
During this time, Rasmussen stole various objects to smuggle back to his time, including a phaser. He also became enamored with Doctor Beverly Crusher, a feeling she did not return. As he attempted to return to the time-pod, the Enterprise-D senior staff confronted him about the thefts in the shuttlebay. When Picard asked to search the craft, Rasmussen attempted to abduct Data by asking that he search the craft. He held him with a phaser, but found it had been deactivated. Picard ordered Worf to arrest Rasmussen despite his pleas to be let go. The pod vanished, stranding him in the future. Worf led Rasmussen to the brig, with Picard welcoming him to the 24th century and remarking there were "more than a few legitimate historians at Starfleet who will be quite eager to meet a human from your era." The Enterprise subsequently deposited Rasmussen at Starbase 214. (TNG: "A Matter of Time")
In 2372, as part of an effort to disprove Worf's claim that the Enterprise-D, unlike Deep Space 9, was immune to security breaches, was false, Odo cited the case of "Berlinghoff Rasmussen, a petty criminal impersonating a scientist, committed numerous acts of theft against the crew of the Enterprise," among others. (DS9: "Bar Association")
Background information Edit
Rasmussen was played by actor Matt Frewer.
The script for "A Matter of Time" described Rasmussen as "a kindly-looking Human male in his late forties [...] wearing a drab, nondescript jump suit and a baggy 'lab' jacket."  According to the script for "Bar Association", his name was pronounced as "BERR-ling-hoff ras-MYU-sen". 
The Star Trek: The Next Generation Companion spells his first name as "Berlingoff".
Rasmussen appeared in the Pocket DS9 novel The Big Game, in which he was released from prison in 2369. He then traveled to Deep Space 9, where he participated in a high stakes poker tournament held at Quark's. The novel also mentioned that Rasmussen claimed to have been on Titanias III during the early 2150s, where he met a Klingon who identified himself as being a member of the House of Duras.
In the Star Trek: Strange New Worlds short story Research, Berlinghoff had the foresight to hide blueprints for the time-travel pod prior to his journey, and was able to retrieve them and successfully rebuild it. He then traveled back to the late-20th century, to the home of his ancestor, J. R. Rasmussen. Together, they travel back to 1964, where they relay information about the future to a television writer named Gene Roddenberry. Rasmussen died sometime before 1999 in an auto accident, while driving to an appointment with producers at Paramount Pictures.
Rasmussen also appeared in the novel Indistinguishable from Magic, where he collaborated with Bok in a scheme to use the NX class Intrepid NX-07 to travel back in time. He did this by pretending to work for a Starfleet team led by Geordi La Forge from the USS Challenger. Bok's plan was unsuccessful and Rasmussen escaped by stealing a shuttlecraft from the Challenger.
In Star Trek Online his character is an obtainable Duty Officer for use in the Duty Officer Assignment system. He is a rare quality Trader and on his card has the quote "Everyone dies! It's just a question of when..."