My guess is that you can do this, but it is a lot of backbreaking work. Even if you decide to 'just' do the demolition, you'll save a bunch of money, but will suffer some sore muscles. The dust factor would occur in either situation - you doing it or having it done.
You would probably want to cut the slab with a saw, then tear out the concrete using a demolition hammer and then dig. If you make a hole first big enough to judge the direction, then you can enlarge it as needed to continue. You may want to replace the line all the way to the main sewer...if it has failed in one place, it is probably on its last legs elswhere.
One of the pros will critique my amatuer thoughts, I'm sure.
Bottom line, especially if you are going to replace the line, doing the demolition and creating the access is the bigger expense. Actually making the connections is the cheap part. In most places, making a connection to the town's sewer main requires a licensed plumber. If you can make it easy for them to lay a new line, you won't need them for long, and the cost should be tolerable.