How to convert a random matrix to Unitary Matrix? I know that a complex matrix $n \times n$ is said to be unitary if $AA^*=A^*A=I$ or equivalently if $A^*=A^{-1}$. But I asked what if there is a random matrix and we want to turn it into an unitary matrix, please also give an example.

If you have access to Matlab or Octave, then producing such matrices is as easy as issuing the following commands:
n = 10;
A = rand(n) + 1i*rand(n);
[U,~] = qr(A);

I recommend that you use software such as this to generate these matrices in general. For small matrices that you would like to work by hand, the Gram-Schmidt procedure is what you want. Here is a simple example involving a real $3\times 3$ matrix. Let
$$
A = 
\begin{bmatrix}
1 & 1 & 1\\
1 & 0 & 2\\
0 & 2 & 1
\end{bmatrix}
$$
We are going to construct a $3\times 3$ unitary matrix $U$ from the columns of $A$. For simplicity I will first construct a matrix $\hat{U}$ with orthogonal columns, and then normalize at the end to get a unitary matrix. The first step is easy, we set the first column of $\hat{U}$ equal to the first column of $A$, i.e., $\hat{u}_1 = a_1$. The second column of $\hat{U}$ is equal to $a_2$ minus any contributions from $\hat{u}_1$, that is, 
$$
\hat{u}_2 = a_2 - \frac{\hat{u}_1\cdot a_2}{\hat{u}_1\cdot \hat{u}_1}\hat{u}_1 = 
\begin{bmatrix}
1\\
0\\
2
\end{bmatrix}
-
\frac{1}{2}
\begin{bmatrix}
1\\
1\\
0
\end{bmatrix}
=
\begin{bmatrix}
\phantom{-}1/2\\
-1/2\\
\phantom{-}2
\end{bmatrix}
$$
The last step is the same as the second except now we must remove from $a_3$ any contributions from $\hat{u}_1$ or $\hat{u}_2$. Thus,
\begin{align}
\hat{u}_3 &= a_3 - \frac{\hat{u}_1\cdot a_3}{\hat{u}_1\cdot \hat{u}_1}\hat{u}_1 - \frac{\hat{u}_2\cdot a_3}{\hat{u}_2\cdot \hat{u}_2}\hat{u}_2\\[1mm]
&=
\begin{bmatrix}
1\\
2\\
1
\end{bmatrix}
- \frac{3}{2}
\begin{bmatrix}
1\\
1\\
0
\end{bmatrix}
- \frac{1}{3}
\begin{bmatrix}
\phantom{-}1/2\\
-1/2\\
\phantom{-}2
\end{bmatrix}\\[1mm]
&=
\begin{bmatrix}
-2/3\\
\phantom{-}2/3\\
\phantom{-}1/3
\end{bmatrix}
\end{align}
The last step to obtain $U$ from $\hat{U}$ is to normalize the columns of $\hat{U}$. Doing so we obtain the unitary matrix
$$
U = 
\begin{bmatrix}
1/\sqrt{2} & \phantom{-}1/(3\sqrt{2}) & -2/3\\
1/\sqrt{2} & -1/(3\sqrt{2})          &\phantom{-}2/3\\
0          & \phantom{-}2\sqrt{2}/3   &\phantom{-}1/3
\end{bmatrix}
$$
As you can see from this simple example, the procedure is quite tedious by hand and is best left to a computer. You can read more about the Gram-Schmidt procedure here.
