Mean
\begin{align*}\widehat{\beta} &= (X’X)^{-1} X’ y \\
&= (X’X)^{-1} X’ (X\beta + u)\\
&{\; \vdots}\\
\widehat{\beta} &= \beta + (X’X)^{-1}X’u\tag{1}\\
\Rightarrow
\mathsf{E}(\widehat{\beta}) &= \beta\end{align*}
In general, estimator $b = \beta + Au$ for any linear estimator
Variance
\begin{align*}
\text{var}(\widehat{\beta}) &= \mathsf{E} \bigl[ { \widehat{\beta} – \mathsf{E}(\widehat{\beta}) } { \widehat{\beta} – \mathsf{E}(\widehat{\beta}) }’ \bigr] \\
&= \mathsf{E} \bigl[ ( \widehat{\beta} – \beta) ( \widehat{\beta} – \beta)’ \bigr]\end{align*}
re-arranging (1)\begin{align*}
(\widehat{\beta} – \beta) &= (X’X)^{-1}X’u\\
\text{var}(\widehat{\beta}) &= \mathsf{E} \bigl[ (X’X)^{-1} X’ u \quad u’X (X’X)^{-1} \bigr] \\
&= (X’X)^{-1} \overbrace{X’ \underbrace{\mathsf{E} \bigl[ uu’ \bigr]}_{\color{red}{1^{st}}\Longleftarrow \sigma^2\text{}} X (X’X)^{-1}}^{\color{red}{2^{nd}} =1} \
%\color{red} \Rightarrow >
\text{var}(\widehat{\beta})\\
&= \sigma^2 (X’X)^{-1}
\end{align*}