Build a plane or rocket for Kerbal high altitudes

TLDR: You need parts you don't have to make a plane that flies that high. But a rockets can do it.

The most efficient way is, of course, to make a high altitude (or space) plane. But you'll need to unlock:

High Altitude Flight

Which cost 300 science. That will unlock:
"Whiplash" Turbojet

Which is capable of high altitude, high speed flight. Slap that behind a plane you'll be able to go beyond 20km without any trouble.

But...

You need to do it with what you've got. I recommend using a solid rocket booster (or 2... or 5) to you get you the altitude and then launch a small rocket powered plane. The flight time won't be as long, but you'll be able to fly a "plane" above 18km.

However...

Trying to do something without the right part is long and difficult path. It will be expensive and time consuming. Your decreased flight time means you probably won't make it to your location. You're probably better to get a new contract and abort that one...

So Pvt. Grichman's answer got me in the right direction. I didn't succeed with his design, though; I always ended up having somehow unstable airplanes. Pretty often, the tail "wiggled" and/or the plane became totally unstable when I fired the rocket boosters.

So I took a slightly different approach:

The main body is a fuel tank with a Swivel engine at the end. I start with the normal jet engines (which takes almost the complete runway). Once it's in the air, it's stable and easy to fly and manages to fly at altitudes between 11k – 12k without problems.

Once I'm near or in the target area, I set the throttle to 0 and start the Swivel. Then I increase throttle until about 2/3 or even full, climbing with a high inclination (it can also go straight up). Immediately starting rocket engines at full throttle turned out to make most of my aircrafts totally unstable, turning them up gradually worked way better.

With this thing, I can easily reach 20k.

I also added a parachute and decoupler since I find it too hard to correctly land this thing yet ;-) I managed to land in the water once, though.

Pvt. Grichmann's answer has been the most reliable way I've found to complete these missions. So long as there's only one high-altitude measurement to take, you should be able to complete a whole set of 3-4 in one flight. Either finish the low-altitudes and then spend all your fuel on the high-altitude boost, or do the boost first and save a little fuel or try to hit the rest through unpowered gliding. Watch out for your SAS if you're running batteries without a jet for a long time.

Another approach, which I was managing with some success before I figured out jets, is to build a rocket and catapult yourself towards the measurement point. With some fins to keep you stable and careful aim, you can hit your desired location and altitude and will have 5-10 seconds to collect data before you hurtle past it and pull your chute. Range with this method is pretty short due to drag.

Third, which I haven't given a good try, would be a carefully-timed deorbit maneuver so you come down through your desired location as you fall from space. Maneuver node plans don't take drag into account, so you'll have to overshoot quite a bit to compensate.