I'll answer two questions, the one you asked and the one you also should have asked. I'll also cover various different scenarios (same subject distance without cropping, same subject distance with cropping, and same framing).
How does a teleconverter affect depth of field?
Let's take a look at this. Depth of field is:
DoF = 2 * x_d^2 * N * C / f^2
f is the focal length,
C is the circle of confusion,
N is the aperture number, and
x_d is the subject distance. If the subject distance stays constant,
and you don't decide that due to less cropping
C should be increased, a doubling of focal length will double the aperture number as well but
C stays constant. Thus, depth of field will be halved by the teleconverter. (If you increase
C due to less cropping needed, the depth of field would stay constant.)
However, sometimes you want to keep equal framing. Then, a doubling of focal length will correspond to a doubling of subject distance. Thus,
x_d^2 / f^2 stays constant and
C stays constant too. However, a doubling of focal length will double
N, and thus, depth of field will be doubled with equal framing.
So, TL;DR: it depends on whether you maintain equal framing by changing subject distance (different DoF), whether you crop (same DoF) or whether you just accept a longer focal length gets you a different picture (different DoF, but in the other direction).
You should also have asked:
How does a teleconverter affect background blur?
This is easier. Background blur disc size (assuming background at infinity) is:
b = f * m_s / N = (f/N) * m_s
The aperture opening,
f/N is maintained by a teleconverter.
m_s is subject magnification, i.e. subject size on the sensor divided by its actual size. If you keep equal framing,
m_s stays constant and thus, with equal framing, background blur disc size is constant.
However, if you don't keep equal framing, the 2x teleconverter doubles
m_s. Thus, you will get more background blur.
But, if you keep the subject distance the same, and have cropped the original image by 2x, and decide that you no longer need cropping due to the teleconverter, then
m_s is doubled by the teleconverter but due to less cropping, the width/height/diagonal of the actually used sensor piece are doubled as well, so blur disc size as a percentage of the actually used sensor piece diagonal stays the same.
So, TL;DR: it depends here again whether you maintain equal framing by changing subject distance (same blur), whether you crop (same blur) or whether you just accept a longer focal length gets you a different picture (different blur).