The coordinate variables are: **v, x, y, u**

As you stated the metric is:

$$ds^2 = g_{i j} dx^i dx^j$$

The matrix has to symmetric and the value of $\delta_{i j}$ depends on **i** and **j** as follows:

If i = j, $\delta_{i j}$ = 1

If $i \neq j$, $\delta_{i j}$ = 0

Now let's find out the value of the matrix's diagonal:

1) $g_{1 1}$

$$ds^2 = g_{1 1} dv^2$$

As this factor does not appear in the given Brinkmann's metric, $g_{1 1} = 0$. Note our first variable is v.

2) $g_{2 2}$

$$ds^2 = g_{2 2} dx^2$$

This factor does appear in the given Brinkmann's metric and $\delta_{i j}$ provides us with the answer, $g_{2 2} = -1$

3) $g_{3 3}$

For the same reason that $g_{2 2} = -1$, $g_{3 3} = -1$

4) $g_{4 4}$

$$ds^2 = g_{4 4} du^2$$

Thus:

$$g_{4 4} = -K_{i j}x^ix^j$$

You can obtain the rest of the matrix proceeding like shown. To illustrate an example different from the diagonal:

5) $g_{1 4}$

$$ds^2 = g_{1 4} dvdu$$

Because of the matrix symmetry we know:

$$g_{1 4} = g_{4 1}$$

As the coefficient of $du dv$ is 1 we know:

$$g_{1 4} + g_{4 1} = 1$$

Therefore:

$$g_{1 4} = g_{4 1} = \frac{1}{2}$$

The entire matrix:

PD: I suggest if you want to learn more about metrics, you check the lectures of Leonard Susskind on youtube, they are great.