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.