Why was Karkaroff eligible for the headmaster position at Durmstrang when he had a bad reputation of being a Death Eater?
Durmstrang definitely had a "horrible" reputation, according to Hermione, within the wizarding community because it did focus more on the Dark Arts, as noted in An Appraisal of Magical Education in Europe. Draco Malfoy bragged on the Hogwarts Express of how his father wanted to send him to Durmstrang, where they actively teach the Dark Arts and not just Defense Against the Dark Arts, which they stick to at Hogwarts. Also according to Malfoy, Durmstrang doesn't admit Muggleborns (I'm not sure about half-bloods). Having an ex-Death Eater as headmaster would seem logical, given that.
‘... Father actually considered sending me to Durmstrang rather than Hogwarts, you know. He knows the Headmaster, you see. Well, you know his opinion of Dumbledore – the man’s such a Mudblood-lover – and Durmstrang doesn’t admit that sort of riff-raff. But Mother didn’t like the idea of me going to school so far away Father says Durmstrang takes a far more sensible line than Hogwarts about the Dark Arts. Durmstrang students actually learn them, not just the defence rubbish we do ...’ - Draco Malfoy
Goblet of Fire - page 147 - Bloomsbury - chapter 11, Aboard the Hogwarts Express
If Durmstrang was an institution that actively discriminated against Muggleborns, having an ex-Death Eater as headmaster wouldn't be seen as problematic, as Death Eaters supported an anti-Muggle agenda. Depending how far into the Dark Arts the school was, having a Death Eater on staff might have been considered an advantage. It would be a stretch to posit Durmastrang didn't know Karkaroff was an ex-Death Eater. Karkaroff's Death Eater activities were public knowledge. I find it impossible to believe that Durmstrang didn't know he was an ex-Death Eater when he was initially hired.
He was eligible because Durmstrang didn't view Death Eaters in the same way that England saw them. From their stance on Muggles/Muggleborns to their stance on the Dark Arts. In fact, his being a Death Eater and well versed in the Dark Arts made him MORE value to teach it.
We have plenty of real world examples.
Snowden is a criminal according to the US yet lives in Russia and is the president of the Freedom of the Press Foundation.
Josef Mengele avoided trail and lived in Buenos Aires, Paraguay and died in Brazil. He hid within the German communities because his money was valuable.
The US government employed Nazis through Operation Paperclip because their knowledge was valuable
Ever heard of Unit 731? They were worse then Mengele. Those scientists were secretly granted immunity in the US for there data. They were valuable.
Karkaroff had knowledge and experience in the Dark Arts. This gave him value. Just like the US didn't have a problem with racism and the science was valuable (or the law vs social values differ in Snowden's case), the moral values are shared and the experience makes that person valuable to the group giving them status anyways.