It's difficult to tell exactly what data is being used, as clubs and the data company like to keep that very secret. It is their intellectual property after all, and different clubs will be looking for different things.
As @fizoxy says, though, you can imagine we look at the number of key metrics that he mentions from percentage of games played by academy graduates and net profit on players recruited, developed and sold, to formations, tactics, style etc.
It's only part of the picture of course, and candidates still need to be good fit for the club. In our specific case, we need to find a manager who supports the fan-owned model the Community Trust. Not sure if we'll have metrics on that, but there could be some.
A number of years ago on one of the Royal Institution Christmas Lectures of all places, I remember the data scientist from Liverpool discussed how they were using certain models. It wasn't related to recruitment, but it was still fascinating.
One of things they had done was to map the pitch into 1 metre squares. Then they took all the match data from their own games, and all the data from other EPL matches, and figured out the chances of a goal coming from possession in each individual square. Then, when they were training, part of their focus was to have tactics and formations that would get the ball into the higher percentage areas as quickly and as often as possible.