$$
\newcommand{\T} {\color{blue}{\text{T}}}
\newcommand{\F} {\color{red}{\text{F}}}
\newcommand{\?} {\color{green}{\text{?}}}$$
This is one way of writing truth tables that comes out pretty nice. Start by writing out the 4 cases for $p$ and $q$:
$$\begin{array} {cccccccccccc}
%
p &\lor &\lnot &(((&\lnot &p &\lor &q) &\implies &q) &\land & p) \\
%
\T & & && & \T & & \T & & \T & & \T \\
%
\T & & && & \T & & \F & & \F & & \T \\
%
\F & & && & \F & & \T & & \T & & \F \\
%
\F & & && & \F & & \F & & \F & & \F \\
%
\end{array}$$
Then start filling in the table 1 operator at a time, starting with the first operator evaluated, the $\lnot$ in the $\lnot p$:
$$\begin{array} {cccc|c|ccccccc}
%
p &\lor &\lnot &(((&\lnot &p &\lor &q) &\implies &q) &\land & p) \\
%
\T & & && \F & \T & & \T & & \T & & \T \\
%
\T & & && \F & \T & & \F & & \F & & \T \\
%
\F & & && \T & \F & & \T & & \T & & \F \\
%
\F & & && \T & \F & & \F & & \F & & \F \\
%
\end{array}$$
Then the value of the $\lor$ in $\lnot p \lor q$, using the values in the $\lnot$ column and the $q$ column:
$$\begin{array} {cccccc|c|ccccc}
%
p &\lor &\lnot &(((&\lnot &p &\lor &q) &\implies &q) &\land & p) \\
%
\T & & && \F & \T & \T & \T & & \T & & \T \\
%
\T & & && \F & \T & \F & \F & & \F & & \T \\
%
\F & & && \T & \F & \T & \T & & \T & & \F \\
%
\F & & && \T & \F & \T & \F & & \F & & \F \\
%
\end{array}$$
Continue filling in for $\implies$ next, then $\land$, then $\lnot$, then $\lor$:
$$\begin{array} {c|c|cccccccccc}
%
p &\lor &\lnot &(((&\lnot &p &\lor &q) &\implies &q) &\land & p) \\
%
\T & \? & \? && \F & \T & \T & \T & \? & \T & \? & \T \\
%
\T & \? & \? && \F & \T & \F & \F & \? & \F & \? & \T \\
%
\F & \? & \? && \T & \F & \T & \T & \? & \T & \? & \F \\
%
\F & \? & \? && \T & \F & \T & \F & \? & \F & \? & \F \\
%
\end{array}$$
If all the values in the final column are true, then the statement is a tautology. If all the values in the final column are false, then it is a contradiction. Otherwise, it could be either (depends on the values of $p$ and $q$, not quite right to say neither).