Depending on the price range you're shopping in:
FIRST - 'cause you can't fix it - Bore condition. Take a flash light, know what to look for. A little darkness, frosting or light pitting never killed anybody but a sewer pipe won't shoot.
SECOND - Head space. You can change this by changing the bolt head but you shouldn't have to get stuck with this on your first try.
THIRD - Metal parts. Is everything there? Right sights for the year etc? The parts can usually be had, but again why stick yourself with a restore right off the hop unless it's stupid cheap?
FOURTH - Furniture condition. Lee Enfield rifles are happiest when "tuned" with a little upwards pressure shimmed into the barrel on the forend. Cracked/shrunk wood will have to be replaced. Cheap to replace for a No.4, if a bit of a pain to color match, not a biggie for a shooter. For a No.1 things get tougher.
FIFTH - Manufactrer/year. Every collector has their pet, and different reasons for them.
I don't know where you're shopping, but know your seller as well. An "as-is-where-is" deal on a questionable L.E. is one to walk away from. There are some nice L.E.s for sale on the EE right now from people who know the gun and aren't out to #$%& you. Dealing with somebody like Tradeex who has the stock and the sterling rep to go with it wouldn't hurt either.
If you have a short list of features you want and know what to avoid, you'll do good. Do some research, check the "milsurp knowledge liabrary" at the top of the Milsurp forum page. Lots of good stuff there.
And post pics when you get your rifle!