> Question1: How to calculate the density function?

\begin{align}
f_{Y_p,Y_{p-1},\ldots,Y_1}(y_p,y_{p-1},\dots,y_1;\mathbf{\theta}) = &\small (2\pi)^{-p/2} \left|\sigma^{-2} \mathbf{V}_p^{-1}\right|^{1/2}\exp{\left[-\frac{(\mathbf{y}_p - \mathbf{\mu}_p)'\mathbf{V}_p^{-1}(\mathbf{y}_p- \mathbf{\mu}_p)}{2 \sigma^2}\right]}\\
=&\small (2\pi)^{-p/2} (\sigma^{-2})^{p/2}\left| \mathbf{V}_p^{-1}\right|^{1/2}\exp{\left[-\frac{(\mathbf{y}_p - \mathbf{\mu}_p)'\mathbf{V}_p^{-1}(\mathbf{y}_p- \mathbf{\mu}_p)}{2 \sigma^2}\right]}
\end{align}

As I said in my comments on this [DSP](http://dsp.stackexchange.com/questions/19182/unable-to-derive-crb-for-ar-model) question, you have an extra term with $\left| V_p^{-1}\right|$ both in your likelihood and loglikelihood.

> Question2: What will be the complete density function and the
> likelihood?

The complete likelihood function is:
\begin{align}
f_{Y_T,Y_{T-1},\ldots,Y_1}(y_T,y_{T-1},\dots,y_1;\mathbf{\theta}) = & f_{Y_p,Y_{p-1},\ldots,Y_1}(y_p,y_{p-1},\dots,y_1;\mathbf{\theta}) \\
& \times \prod_{t=p+1}^T f_{Y_t|Y_{t-1},\ldots,Y_{t-p}}(y_t|y_{t-1},\dots,y_{t-p};\mathbf{\theta})
\end{align}

And the loglikelihood: 

\begin{align}
{\large\mathcal{L}}(\theta) = &\log f_{Y_T,Y_{T-1},\ldots,Y_1}(y_T,y_{T-1},\dots,y_1;\mathbf{\theta})\\
 = &\log f_{Y_p,Y_{p-1},\ldots,Y_1}(y_p,y_{p-1},\dots,y_1;\mathbf{\theta})  +\log \prod_{t=p+1}^T f_{Y_t|Y_{t-1},\ldots,Y_{t-p}}(y_t|y_{t-1},\dots,y_{t-p};\mathbf{\theta})\\
=&-\frac{p}{2}\log(2\pi)-\frac{p}{2}\log(\sigma^2)+\frac{1}{2}\log\left| V_p^{-1}\right|-\frac{1}{2\sigma^2}(\mathbf{y_p} - \mathbf{\mu_p})'\mathbf{V}_p^{-1}(\mathbf{y_p}- \mathbf{\mu_p})\\
&-\frac{T-p}{2}\log(2\pi)-\frac{T-p}{2}\log(\sigma^2)\\
&-\frac{1}{2\sigma^2}\displaystyle\sum_{t=p+1}^T\left(y_t
 - c - \phi_1 y_{t-1}- \phi_2 y_{t-2}-\cdots- \phi_p y_{t-p}\right)^2\end{align}