It doesn't look, from the pictures, that its sloping toward the house. If it is, it isn't significant.
What is the issue? Water in the basement?
I wouldn't go to all the trouble of a french drain just yet. Just build a swale. A swale is a low spot. build it close to the property line and extend it so water can run out to the street or far away from the house. This could be done with just a shovel (and a sledge to break up the concrete path). If you wanted to go all he man you could maybe rent a small bobcat for a couple of hours.
Remember call before you dig. You have a gas line that may be in the way.
If it were me - I would combine the grade fix with a new walkway. Maybe some nice pavers. You need to do grading work for that anyway. If you can extend your downspouts underground, with solid pipe, to daylight. You will also do wonders.
If you want to get some prices - Talk to a landscaper, not a water mitigation guy, I'll bet you beat the prices you have significantly and also get something nicer.