I'm originally from Pittsburgh and... no. Not really. Not close to the city.
When I visited my parents a while ago I hiked the Laurel Highlands Hiking Trail, which is quite nice, but a bit far from the city. One terminus is Ohiopyle, and outfitters there offer shuttles. I enjoyed it a lot, but that area was practically my second home as a kid, so I may be biased. It's about 70 miles, with many options for chopping it into smaller pieces. It might work great for four days.
Most decent hiking is in the center of the commonwealth like that.
Black Forest has been mentioned (ninja'd- curse you, John!). There's also Black Moshannon State Park, which has big loop hike that is doable in a weekend, called the Allegheny Front Trail. It's 42 miles. It's close to State Park, so you might be mingling with college kids. There are a bunch of trails in Allegheny National Forest, and I think there's even a guidebook, but that's quite a ways from the city. The North Country Trail is 95 miles, but it's so far from the city that part of it is in New York.
Some that I've heard of but never done are:
The Baker Trail, from the northern suburbs to Allegheny National Forest, 132 miles. There are shelters. And a guidebook.
The Warrior trail is 67 miles from Greenboro into West Virginia. I know essentially nothing else about it.
There's the Great Allegheny Passage, a rails-to-trails system that connects to the C&O towpath route in Maryland all the way to Washington D.C. Biking the whole route is on my life list for someday. It's not wilderness near Pittsburgh, though- and there are a lot of bicyclists. It passes within a literal stone's throw of my parent's house, though. It's actually rather nice, and there are companies that will drop you off and pick you up anywhere along it that you like, and rent you a bike.
I haven't lived there in decades, though, so my info is probably dated. Hopefully others will speak up.