Katarina lived in ancient Troy at the time of the famous war as a handmaiden to the princess Cassandra. The Doctor, Vicki and Steven met her in the last episode of The Myth Makers, when all hell was breaking loose. She made friends with Vicki, who sent her to help the Doctor get a wounded Steven back to the TARDIS. Katarina came aboard and the Doctor promptly absconded with her, without asking if she'd like to come along. (Most of the early companions were either stowaways or kidnap victims.)
With Vicki having stayed behind in ancient Troy, Katarina served as her replacement aboard ship. When the TARDIS landed on the planet Kembel at the start of The Daleks' Master Plan, Katarina stayed in the ship to tend to the still-wounded Steven while the Doctor set off to look for help.
The TARDIS is found by Space Agent Bret "not the Brigadier" Vyon, who demands that Katarina fly him off the planet. Operating a time machine wasn't covered in Trojan handmaiden school, so Katarina is somewhat at a loss. Luckily Steven comes to long enough to knock Bret out before passing out again.
But Bret's not all bad, and, with a promise that he will help Steven, Katarina agrees to let him free from the Doctor's restraining chair (don't ask). Bret cures Steven and helps the the duo escape from the Daleks who have surrounded the TARDIS.
Isn't Bret dashing? The three of them meet up with the Doctor, who's surprised that Katarina let Bret go free.
Katarina convinces the Doctor that Bret's on the side of the angels, and the four set off to foil the Daleks' evil scheme of the day.
They steal the rare element taranium from the Daleks and flee the planet in a convenient spaceship. Unfortunately, the Daleks shoot them down, and they are forced to land on a prison planet. They manage to take off again, but not before one of the convicts, Kirksen, sneaks aboard via the ship's airlock. When Katarina is sent to check the airlock - because why wouldn't you send the girl from ancient Troy to check the airlock? - Kirksen grabs her.
Kirksen demands to be taken to the nearest planet, which just happens to be the planet where the Daleks are waiting to kill all of them. The Doctor and Steven agree in order to save Katarina's life, but she reaches for the airlock controls...
...and ejects herself and Kirksen into space, sacrificing herself to save the Doctor, Steven and Bret from the Daleks. Let's take another look at that picture, shall we?
Not horrifying at all. Poor, poor Katarina. With one push of a button she became both the first Doctor Who companion to die, and the shortest lived companion from the original run. Although she spanned two stories, she only appeared in five episodes total, and she only really appeared fully in three - we met her at the end of the last episode of The Myth Makers, and she became vacuum food at the start of the fourth episode of The Daleks' Master Plan. Kamelion the shape-changing robot got more airtime than that, and he didn't even work. (More on him later. Much, much later.)
Katarina handled her travels in the TARDIS (ok, travel - she only made one stop) remarkably well - perhaps to keep herself from instantly dying from culture shock, she decided that the Doctor must be a god (Russell T Davies would later take this idea and run with it). He was Zeus, she rationalized, and the TARDIS was his flying temple. She was already dead, and the TARDIS was her conveyance to the afterlife. The Doctor protested, and although she stopped calling him a god at his insistence, she never stopped believing it.
Although Adrienne Hill was signed on to be the new "Doctor Who girl", the writers very quickly realized that they didn't know what to do with the character. They felt her naive world view would make for limited story-telling options, and so they killed her off, replacing her (in the short term) with Space Security Agent Sara Kingdom (more on her next time).
It's a shame, really, because Katarina had a lot more potential than the writers gave her credit for. She's often described as naive, but that doesn't seem quite fair - she showed good judgement in defying the Doctor's orders by releasing Bret to save Steven - which must not have been easy for her, since she would have believed she was defying her god. And she clearly isn't stupid - in her final scene, she figures out which button to push to jettison herself out of the airlock, despite barely understanding what an airlock was (Steven's horrified cries of "No, not that one!" may have helped, serving as a morbid game of hot-and-cold).
Keeping Katarina aboard the TARDIS could have helped the writers, not hindered them, if they had so chosen. She could have easily served as the stock question-asker to move the plot along ("What are you doing, Doctor? What does that mean? What does this button do?" Maybe not that last one.), with Steven as a more experienced counterbalance. Her journey from bewildered innocent to worldly traveler could have made for great stories - just a year later they'd do exactly that with 18th century Scottish piper Jamie McCrimmon - but the writers' weren't interested in that kind of character development for a companion at this time, so out the airlock she went.
In the expanded universe of Doctor Who, though, leaving the TARDIS is never the end. Companions are constantly brought back in tie-in novels, comics and audio adventures, with new stories taking place in the gaps between the original television adventures. Katarina has presented a problem for writers of these licensed spin-offs, though - there are no gaps in her appearances in which to fit new stories, since the end of her first story leads directly in to the start of her last, and her death makes it a little difficult to write about her life after leaving the Doctor's company. ("Katarina froze. Her body floated a bit, then got caught in a planet's gravity well. She burned up on re-entry. The end.")
But Doctor Who fans are nothing if not creative, and Katarina has managed to make a handful of post-death appearances. The Seventh Doctor met her as a child in a comic in Doctor Who Magazine (time travel, you know), and also encountered her as an adult, inexplicably alive on a far-off world.
Spoiler! She was really a shape-changing alien. Ah, well, nice to see her again anyway. Short stories have shown her in the Greek underworld trying to make it to the Elysian Fields, and in an alternate timeline having all sort of adventures with the Doctor and Steven.
Katarina - potential squandered, all but one episode lost perhaps forever, only remembered for being the first to die. And not even remembered well - Adric's really the go-to dead companion when you want to dredge up some of the Doctor's guilt. What's a handmaiden gotta do to get some respect?