Let me change your notations slightly: you work in three dimensions and you want to compute
$$
u_{jk}(x)=\int e^{2i\pi x\cdot \xi} \frac{\xi_j\xi_k}{\vert \xi\vert^4} d\xi,
$$
where the integral should not be taken as an $\dots$ integral. You have by homogeneity (in $n$ dimensions the Fourier transform of an homogeneous distribution with degree $\lambda$ is homogeneous with degree $-\lambda-n$) and "radiality"(the Fourier transform of a radial function is also radial)
$$
u_{jk}=D_{x_j}D_{x_k}\int e^{2i\pi x\cdot \xi} \frac{1}{\vert \xi\vert^4} d\xi=cD_{x_j}D_{x_k}
\vert x\vert ,
$$
where $c$ is a constant and $D_t=\partial_{t}/2i\pi$. We get
$$
j\not=k,\quad u_{jk}=c_1\partial_{x_j}\vert x\vert^{-1}x_k=-c_1 x_jx_k\vert x\vert^{-3},
$$
$$
u_{jj}=c_1\partial_{x_j}\vert x\vert^{-1}x_j=c_1 \bigl(\vert x\vert^{-1}
-x_j^2\vert x\vert^{-3}\bigr).
$$