Capacity of binary not symmetrical channel
You need to compute $H(Y)$ by its definition:

$$ H(Y) \triangleq \Pr(Y=0) \log \frac{1}{\Pr(Y=0)} + \Pr(Y=1)\log\frac{1}{\Pr(Y=1)}$$

To be able to compute this formula, you need to figure out what is $\Pr(Y=1)$ and $\Pr(Y=0)$. These two measures actually depend on the distribution of $X$. 

To compute $H(X|Y)$ again, use the definition.
$$H(Y|X) \triangleq \Pr(X=0) H(Y| X=0) + \Pr(X=1)H(Y|X=1)$$

where $H(Y | X=0)$ is easy to compute from the original entropy definition $H(Z)$, for a random variable $Z$ that is distributed same as $(Y|X=0)$. This is immediately given by your $P(Y|X)$.

In order to compute the channel's capacity, you will need to find the distribution $p(x)$ that maximizes $H(Y)-H(Y|X)$. 



Maybe you should also check a textbook about information and entropy. For instance, try Cover &amp; Thomas, "Elements of Information Theory", Chapter 7 (Channel Capacity).