It seems this would be an economical way to explore anything beyond the heliopause, given that the probe would carry limited fuel.
As Local Fluff mentioned, you'd have to match velocities with the comet for a non destructive rendezvous. And when you match velocities you're already on the same trajectory as a comet.
But a comet might give radiation shielding, water, and other consumables -- a lot of mass that would be hard to launch from earth.
The Heart of the Comet by David Brin and Gregory Benford looks at landing humans on Halley's Comet. This was one of Brin's early novels.
Before he became a science fiction writer, Brin was a competent scientist (in my opinion). His doctoral dissertation was on cometary evolution. He argued that if perihelion outgassing wasn't too violent, a comet might accumulate an insulating mantle that would preserve an icey core. He speculated that many of the dead comets that no longer outgas might still have icey cores. So I believe Brin was well qualified to write a story set on a comet. I found the yarn entertaining.
Manned rendezvous with a long period comet would be very difficult. When in earth's neighborhood comets are moving at a pretty good clip so a soft landing would take a lot of delta V. As a long period comet heads back out into the outer solar system, passengers would have to endure many decades or even centuries of extreme cold and very little sunlight. So they'd have to have a robust and reliable nuclear power source. This sort of scenario is pretty much science fiction.
However a probe rendezvous with a short period comet is not science fiction. Rosetta is due to rendezvous with comet 67P/Churyumov-Gerasimenko this August.
In order to connect with the comet, you need to get yourself the same orbit as the comet has. And you could do that without any comet. That's the basic flaw in your idea there.
If you want the comet to radically change the speed of your probe, then you need a technology to absorb that "collision" between your probe and the comet. If a high speed train passes you by at 360 km/h that is just 0.1 km/s. Imagine jumping abord it on the fly! That's a challenge. But in order to get any speed of relevance for interplanetary space travel, you need to 100 fold that speed difference. What structure could survive such a speed change?
Still, NASA has very recently funded a one man one year investigation into exactly this scenario (if only there were long tethers made out of nanotubes) :-D