Assuming the closest is even viable, which it probably isn't, 4.2 Light Years = 39735067984839.36 Kilometers. The fastest thing (only thing) man has sent out of our solar system is Voyager 1, which at its current speed, if it was pointed in the right direction would take about 73,775 years to reach the target. Considering you probably don't want to run into it at that speed, you will have to accelerate and decelerate. Which it doesn't have the fuel for (never mind its RTG energy source is only good for 60-80 years), but even if it did would roughly double the time to reach the intended target to about 147,550 years. OK well that's not quite true, it would only add an insignificant amount of time because not a lot of time was actually spent to accelerate in the first place. However in the example below where you do not coast for tens of thousands of years, and accelerate til the midway point and then immediately start to decelerate it would double whatever you speed VS distance is anyway.
Sure you could accelerate and decelerate much harder than that to get there much faster, approaching whatever value of c is currently capable at launch. However by any measure, unless some magic energy source and method of propulsion is devised, the required energy at least at today's standards would require carting around the hydrogen energy mass of our sun for the trip. Some other methods of insitu material gathering such as ram scoops picking up interstellar dust are as likely as the fiction, as again unless some dark matter type thing which is everywhere (presumably) is harnessed, the amount of mass available is pretty low, space as it turns out is pretty damn empty.
Not to mention the weirdness of relative time as one approaches c on a ship compared to Earth, as while it may take less than the 75k years voyager would, here on Earth many more years will have elapsed. As to how many, I have no idea, that is beyond my math calculating ability (as is generally most of what I have currently written I am sure will be pointed out).
Never mind trying to maintain a ship, machinery, technology, or even a society that long!
More likely colonization will involve self replicating and regenerating robotic ship carrying a genetic payload and an informational database (likely with a terra forming mission proceeding it). Which would be more like favorable seeding for similar evolution and life to occur, than an actual "colony". Then again, that would also require pretty adaptive programming and AI, which would likely mean we would probably be fertilizer for our robotic overlords petunia plants.
So I guess I am saying as a thought experiment it is sort of interesting, but at this point (or any really foreseeable point in our future), it is all a bit far fetched by even the loosest standards.