Barring having a very large power to send information back on a very large angle, it seems to me that the level of precision to send this information back so that it can be captured by an antenna on Earth is prohibitively hard to get. How is NASA accomplishing it?
The high gain antenna of New Horizon as an opening angle of its beam of about 0.6°. That means, it has to be pointed at Earth with an error margin of 0.3°. As a practical example, this is more like pointing a torch (flashlight) with a (well focused) beam at a far target than aiming with a tiny Laser spot. For comparison, 0.6° is slightly larger than the apparent size of the Moon in our skies which is 0.5° across.
The antenna is fixed to the spacecraft, so that the whole space probe has to rotate (that's the reason there was no direct data download during the encounter with Pluto). Rotating is accomplished by its on-board thrusters that can be used to adjust rotation very precisely.
Now the only remaining point is to figure out were Earth is located. Luckily somebody installed a bright beacon light relatively close to Earth (better known as the Sun) that can be used to find it. From Pluto Earth is at most 1.3° off to either side of the Sun.
New Horizons is equipped with a star tracker - essentially a camera that takes images of the sky and some software that reads the position and brightness of stars and compares them with a map. If, for any reason, pointing the antenna at Earth fails the space probe can switch to its smaller medium gain antenna which can work even when missing Earth by up to 10°. Essentially this allows to operate in a pure Sun-tracking mode without knowing the precise orientation of the probe.
And, as a last backup, there is also an omnidirectional antenna that can receive commands under almost any conditions to help getting the spacecraft operational again. Due to its low gain, this antenna could be used during the initial phase of the flight only - after traveling more than a few AU the signal received is too weak to be useful.