This has bugged me for some time.
Tycho Brahe's data on planetary observations, presumably, consisted of the direction in which a planet was observed at a given date and time, but not the distance to the planet. What techniques did Kepler use to add a depth dimension to these observations, to create the three-dimensional data that one can start studying to arrive at his three laws?
At the time of Kepler distances were not measured or observed directly. Or if measured, the results were completely wrong. All his laws are stated in terms of RATIOS of distances, and these ratios can be in principle measured from the geometry of the situation. When you describe everything as seen from the Earth (as the ancients did), distances are completely irrelevant. But when one uses heliocentric system, all ratios of distances to the distance from Earth to Sun can be obtained from angular observation.
Of course this is only the general principle. The details are MUCH more complicated.
EDIT. But the idea is the following: suppose for simplicity that everything happens in the same plane, and that the planets move on circles. (This is actually a good approximation because the inclinations of the orbits are small and excentricities are also small. As seen from the Earth, the Sun rotates on a circle of radius $r$ uniformly, with center at the Earth. The planet rotates around the Sun on a circle of radius $R$, also uniformly. Suppose that at three different times you observe the direction on the planet, this essentially means that you measure two angles on your picture. And two other angles you know because you know the time of observations and Sun's rotation speed. From these angles by pure geometry you can find the ratio $r/R$. Just make a picture, and consider this a high school geometry exercise (of moderate difficulty). This was already known to Ptolemy, by the way.
These ratios were known to Kepler, and by playing with them he discovered his 3-d
law. But his greatest achievement is the FIRST law: he was able to derive from
the observations (of the same kind that I described) that the planets really
do not move on circles but on ellipses. How exactly he did this I cannot explain within the space allowed here:-) But his own explanation is available in English btw.
This is not really a complete answer, but is too long to be a comment. Alexandre Eremenko has written a nice answer, which this is meant to supplement.
One point to understand is that what Kepler did was an exercise in curve-fitting. There are 6 parameters needed in order to describe a Keplerian orbit. (You can tell there are 6 by counting degrees of freedom. An initial position vector and an initial momentum vector are enough to define such an orbit.) When you observe an object's position on the sky, in principle you need 6 numbers in order to define its orbital elements. For example, you could do this by finding its declination and right ascension on three different nights.
All of this only works if you assume a Keplerian orbit. For example, suppose that the sun violated the laws of physics, as currently understood, by moving randomly along the line connecting the sun to the earth. This would have no effect at all on the sun's right ascension and declination, so we could never detect this motion from measurements of those coordinates.
Another thing that may help with intuition is to realize that this is directly analogous to the parallax method for measuring the distances to stars. The only difference is that we take a star to be distant and at rest, which is a degenerate case of Keplerian motion.
What techniques did Kepler use to add a depth dimension to these observations, to create the three-dimensional data that one can start studying to arrive at his three laws?
So I think this part of the question makes an incorrect assumption, which is that Kepler first found the three-dimensional motion, and then inferred Kepler's laws from it. The three-dimensional motion was already known and modeled using epicycles, with the earth arbitrarily taken to be at rest. Kepler was refining this previously developed 3-d model and also (trivially, from a mathematical point of view) changing the origin of coordinates.
