Will this series converge, will its limit exist (and if so, what is it), and will its sum be $> 0$ or $< 0$? The series : $$\lim\limits_{n \to  \infty }  \sum\limits_{k=1}^n - (-1)^{k} \ln (k)$$
We know that, for a sum $\sum\limits_{k=1}^\infty a_k$ to converge, a necessary condition is that : 
$$\lim_{k\to\infty} a_k = 0$$
We can write the above series as 
$$\ln(1)-\ln(2)+\ln(3)-\ln(4)+\ln(5)-\ln(6)\cdots\infty$$
It becomes : \begin{align}
& =\ln\left(\frac{1}{2}\right) + \ln\left(\frac{3}{4}\right) + \ln\left(\frac{5}{6}\right) + \ln\left(\frac{7}{8}\right) + \ln\left(\frac{9}{10}\right) + \cdots\ln\left(\bf{\frac{n}{n+1}}\right)\\
\end{align}
and $$\text{as,}\quad n\to\infty\qquad\qquad\lim\limits_{n\to\infty}\left(\bf{\frac{n}{n+1}}\right) = 1$$ 
thus, $$\lim\limits_{n \to \infty}\bf{ln}\left(\bf{\frac{n}{n+1}}\right)=0\text{,}$$
that is, $$\lim\limits_{k\to\infty} a_k = 0\text{,}$$ 
So it should converge, we also observe that all the terms inside $log$ are less than 1 so all the individual $\textit{log }$ values would be negative and thus making the sum of series negative.
We also observe that as $n\to\infty$ the values inside $log$ will tend to $0$ from negative side(because each individual $log$ value is negative)thus, the absolute value of individual $log$ values would slowly decrease, with the greatest absolute value of $ln\left(\bf{\frac{1}{2}}\right)$
But we can write the above series in another way :
\begin{align}
& = \ln(1)-ln(2)+\ln(3)-\ln(4)+\ln(5)-\ln(6)\cdots\infty\\
& = \qquad-\ln(2)+\ln(3)-\ln(4)+\ln(5)-\ln(6)\cdots\infty\qquad\qquad\{\ln(1)=0\}
\end{align}
It becomes :
$$\ln\left(\frac{3}{2}\right) + \ln\left(\frac{5}{4}\right) + \ln\left(\frac{7}{6}\right) + \ln\left(\frac{9}{8}\right) + \ln\left(\frac{11}{10}\right) + \cdots\ln\left(\bf{\frac{n+1}{n}}\right)$$
Again,
$$\text{as,}\quad n\to\infty\qquad\qquad\lim\limits_{n\to\infty}\left(\bf{\frac{n+1}{n}}\right) = 1$$ 
thus, $$\lim\limits_{n \to \infty}\bf{ln}\left(\bf{\frac{n+1}{n}}\right)=0\text{,}$$
that is, $$\lim\limits_{k\to\infty} a_k = 0\text{,}$$ 
So it should converge, we also observe that all the terms inside $log$ are greater than 1 so all the individual $\textit{log }$ values would be positive and thus making the sum of series positive.
Now this is where it CONTRADICTS our above statement that the sum of series would be negative
We also observe that as $n\to\infty$ the values inside $log$ will tend to $0$ from positive side(because each individual $log$ value is positive)thus, the absolute value of individual $log$ values would slowly decrease, with the greatest absolute value of $ln\left(\bf{\frac{3}{2}}\right)$
One may further argue that in our original expression:
$$\lim\limits_{n \to  \infty }  \sum\limits_{k=1}^n - (-1)^{k} \ln (k)$$
the series does not Converge as our necessary condition,
$$\lim\limits_{k\to\infty} a_k = 0$$ 
does not satisfy because $- (-1)^{n} \ln (n)$ tends to $\infty$ or $-\infty$ depending on the value of $'n'$ as $n\to\infty$, thus contradicting our another statement that : 'this series converges'
Now at last, how do we calculate the limit of this series (even an approximate value).
Edit : What would be the answer to the same question if the series would have been:
$$\lim\limits_{n \to  \infty }  \sum\limits_{k=1}^n - (-1)^{k} \ln \left(\frac{1}{k}\right)$$

The answer is simple: It does not converge. It is similar to the old paradox
$$
S = 1-1+1-1+1 \cdots
$$
can be written as 
$$
S=(1-1)+(1-1)+\cdots = 0
$$
but also as
$$
S= 1-(1-1)-(1-1)\cdots = 1
$$
But if you avoid these dubious operations you see that $S$ actually oscillates between $0$ and $1$ and never converges. Your function is similar: $\ln(k)$ grows for each $k$ while $(-1)^k$ makes it oscillate more and more wildly as $k$ increases. 
As you mentioned yourself, we need $a_k \rightarrow 0$. But clearly $(-1)^k\ln(k)$ does not converge to $0$ (its limit does not even exist, as it keeps going up and down in progressively larger swings).
