How much time do you have and what will your budget be?
What are your skills and commitments "back home"? Would you have anything to offer charitable/volunteer projects? I have issues with gap-year volunteers who have few skills to offer and pay somebody else to organise their year-out for them, but I do like the idea of identifying volunteer projects and joining them up as a basis of a trip.
A book I found myself nodding, chuckling and shaking my head at was Dark Star Safari by Paul Theroux. Egypt to Cape Town overland. Caution: may contain spoilers for your own trip. By which I mean he seems to have had a good time and met lots of interesting people. Your mileage may vary.
I reckon scotsman has it about right: too much unpleasantness in the west and centre. I know people doing business in DRC and Central African Republic and nothing happens without greasing the wheels.
fontmoss might be onto a good one: I commute between Francistown and Maun (Botswana) every couple of weeks and the overland trucks are endless. As far as I can tell most of them start in South Africa and don't go much further north than Maun/Victoria Falls. Typically the ones that do the whole north-south/south-north trip are UK registered - you can search for things like africa overland, cape to cairo, etc. for companies. I doubt the drivers would take you on informally as a helper for the whole trip but it might work out for short stretches.
However, ask yourself what kind of trip you want: I get the feeling that sitting on an overland truck driving between campsites (mostly on tar roads) leaves you a bit isolated from the environment. A bus-truck full of sunburned firstworlders dozing or reading novels and getting silly-drunk in the evenings wouldn't be on my list of things to do.
Also, you might find yourself on a bit of a mad dash from one place to the next to finish the journey in time. If it was me I wouldn't set myself the goal of "Cape to Cairo," but rather just meander around and see how far I could get within my budget of time and money.
The majority of people are genuinely friendly and you can have great conversations, but the ones that believe in wealth redistribution and free enterprise will be on you like iron filings on a magnet. My advice: travel light and low-tech.