Draw a spiral like an ellipse

Your code shows a circular spiral because you punch in a circular spiral. Try e.g.

\documentclass{article}
\usepackage{tikz}
\begin{document}

    \begin{tikzpicture}
        \draw [rotate=-45]
              plot[domain=0:25.1327,variable=\t,smooth,samples=75] ({\t r}: {0.002*\t*\t} and {0.004*\t*\t} );
    \end{tikzpicture}

\end{document}

And to add the axes and the arrow, you could use

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{decorations.markings}
\begin{document}

    \begin{tikzpicture}
        \draw[-stealth] (-3,0) -- (3,0) node[below]{$x$};
        \draw[-stealth] (0,-3) -- (0,3) node[left]{$y$};
        \draw [rotate=-45,postaction=decorate,decoration={markings,
        mark=at position 0.93 with {\arrow{latex}}}]
              plot[domain=0:32,variable=\t,smooth,samples=75] ({\t r}: {0.002*\t*\t} and {0.004*\t*\t} );
    \end{tikzpicture}

\end{document}

A PSTricks solution only for comparison and future purposes.

\documentclass[pstricks,border=12pt,12pt]{standalone}
\usepackage{pst-plot}
\begin{document}
\pspicture[algebraic,plotpoints=500](-4,-4)(4,4)
    \rput{-45}(0,0){\psparametricplot[arrows=->]{0}{Pi 15 mul}{t^2*cos(t-Pi/3)/1000|2*t^2*sin(t-Pi/3)/1000}}
\endpspicture
\end{document}