Using the kinds of propulsion systems we have, and taking advantage of a 'gravitational slingshot' from Jupiter, we could probably get up to 150,000 miles per hour. The Galileo probe managed to get to about 106,000 miles per hour and currently holds the record for the fastest speed ever achieved by an artificial body.
The nearest star is Proxima Centauri at a distance of 4.2 light years. At a speed of 150,000 miles per hour, it would take about 17,900 years to reach this star. At that point, you would then have to figure out some way to loose a lot of your velocity, or you would fly right past this star into the depths of interstellar space.