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.