Just bought in LP last year. Get a good inspector. Like any other home, evaluate the age of roof, windows, and appliances.

Observe the adjacent units. Talk to people on the block - you want to get a feel for whether you'll be able to stand the neighbors. Visit and evaluate the noise levels at morning, midday and night. If it has a skylight, pay close attention to how well it was installed - they're almost always the first thing to leak. Check for water infiltration along the edges of the top floor and the basement/crawlspace. Check the alley if you have one - will it be easy to navigate and/or bring your trash out? Does it seem like a popular cut-through or loitering spot? Check around commute times to see the parking/traffic situation.

Take a look at the brick if that's part of the structure - it should be pretty obvious if it's deteriorating. If the brick is bowing out or looks swollen, run away. run all the sinks, flush the toilets, make sure that the water pressure and drainage is good and doesn't knock or anything. Look for any signs of vermin, especially if there is a restaurant on your block.