Properties of Power Spectrum Density
So is that value always one as it's a delta also


What? no. A dirac delta doesn't have a "value". It has an integral area, but not a value at the point where it's not 0. 

The area of the dirac is the power of the DC component, i.e. P=U²/R.


  how can i calculate the DC and AC power 


Strange question. Remember, this is the DC component. Not AC. 

To calculate the power, assuming your PSD is called \$S(f)\$:

\$P= \lim\limits_{\epsilon\rightarrow 0} \int\limits_{-\epsilon}^\epsilon S(f) \, df \$


  and if i passed that Random process through a channel of specific function, how can i get the output ?


Well, if your transmit signal is \$s(t)\$, and the channel is \$h(t)\$, then the receive signal is \$r(t) = h(t) * s(t)\$ with \$*\$ being the convolution operator. You should have learned that before someone explained stochastic signals and PSD to you. 

Due to the properties of the Fourier transform, the FT of the convolution of the time domain signals/systems is the product of their transforms, i.e.

\$ R(f) = H(f) S(f) \$. 

Apply above integral to this.