How can I prove: $\sum_{n\leq x}\frac{\phi(n)}{n^2} = \frac{\log x}{\zeta(2)}+\frac{C}{\zeta(2)} + A + O\left(\frac{\log x}{x}\right)$ The problem is that prove that $$\sum_{n\leq x}\frac{\phi(n)}{n^2} = \frac{\log x}{\zeta(2)}+\frac{C}{\zeta(2)} + A + O\left(\frac{\log x}{x}\right)$$ 
where $C$ is Euler's constant and $A = \sum_{n \geq 1}\frac{\mu(n)\log n}{n^2}$

The following is things I did try:
$$\sum_{n\leq x}\frac{\phi(n)}{n^2} = \sum_{n\leq x}\frac{1}{n}\frac{\phi(n)}{n} =\\ 
\sum_{n\leq x}\frac{1}{n}\sum_{d\mid n} \frac{\mu(d)}{d} = \sum_{q\leq x}\frac{1}{q} \sum_{d\leq x} \frac{x}{q}\frac{\phi(d)}{d^2} =\\
\sum_{q\leq x}\frac{1}{q}\left( \frac{1}{\zeta(2)} + O\left(\frac{q}{x}\right) \right) = \frac{\log x}{\zeta(2)} + \frac{C}{\zeta(2)}  + O\left(\frac{1}{x}\right) + \sum_{q\leq x}\frac{1}{q} O\left(\frac{q}{x}\right) $$
Here $\mu$ is the Möbius function and $\phi$ is the Euler totient function.

This is to obtain the estimate OP wanted. We have in fact ($+A$ should be corrected to $-A$):
$$
\sum_{n\leq x }\frac{\phi(n)}{n^2}= \frac6{\pi^2}(\log x+C) -A +O\left(\frac{\log x}x\right).
$$
The OP had a good start indeed:
$$
\begin{align}
\sum_{n\leq x}\frac{1}{n}\sum_{d\mid n} \frac{\mu(d)}{d}&=\sum_{d\leq x} \sum_{n\leq x, \ d|n} \frac{1}{n} \frac{\mu(d)}{d}\\
&=\sum_{d\leq x}\sum_{q\leq \frac xd} \frac1{dq} \frac{\mu(d)}d \ \ \ \textrm{(Substitution $n=dq$)}\\
&=\sum_{d\leq x}\frac{\mu(d)}{d^2} \sum_{q\leq \frac xd} \frac 1q \\
&=\sum_{d\leq x}\frac{\mu(d)}{d^2} \left( \log\frac xd + C + O\left(\frac dx\right)\right)\\
&=\sum_{d\leq x}\frac{\mu(d)}{d^2}( \log x + C) - \sum_{d\leq x}\frac{\mu(d)\log d}{d^2} + O\left(\frac{\log x}x\right)\\
&=\frac{6}{\pi^2}(\log x+ C)-A +O\left(\frac{\log x}x\right) \ \ \ \textrm{(Replacing $\sum_{d\leq x}$ by $\sum_{d=1}^{\infty}$ contributes $O\left(\frac{\log x}x\right)$)}
\end{align}
$$
