Thanks go to Andrew, J. M., and David Speyer! The following solution leans heavily on what these three have already posted.
(In the interest of having a complete solution I've added the argument that gets from the OP's sum to Andrew's reformulation of it.)
Part 1: Getting to Andrew's gamma function formula.
Since $$\int_0^1\sum_{m=1}^{k-1}\frac1{x+m}\, dx = \sum_{m=1}^{k-1}(\log(m+1) - \log m) = \log k,$$
we can rewrite the original formula as
$$\sum_{k=1}^n \binom{n}{k}(-1)^k \log k=
\sum_{k=2}^n \binom{n}{k}(-1)^k\int_0^1\sum_{m=1}^{k-1}\frac1{x+m}\, dx = \int_0^1 \sum_{m=1}^{n-1} \frac1{x+m} \sum_{k=m+1}^n \binom{n}{k}(-1)^k\, dx.$$
Since alternating row sums of binomial coefficients are easy to evaluate (see, for instance, Concrete Mathematics, Identity 5.16), this becomes (and then switching the index back to $k$)
$$\int_0^1 \sum_{m=1}^{n-1} \frac1{x+m} (-1)^{m-1} \binom{n-1}{m}\, dx = \int_0^1 \sum _{k=1}^{n-1}\binom{n-1}{k} \frac{(-1)^{k-1}}{k+x} \, dx$$
$$ = \int_0^1 \left(\frac{1}{x} - \sum _{k=0}^{n-1}\binom{n-1}{k} \frac{(-1)^k}{k+x}\right) \, dx.$$
The remaining binomial sum is actually the partial fractions decomposition of $\frac{(n-1)!}{x(x+1)\cdots (x+n-1)}$. (This is identity 5.41 in Concrete Mathematics. From another perspective - also discussed in Concrete Mathematics - the binomial sum is $(-1)^n$ times the $n-1$ difference of $\frac{1}{x} = (x-1)^{\underline{-1}}$. Applying the finite difference rule $\Delta x^{\underline{m}} = m x^{\underline{m-1}}$ successively $n-1$ times thus gets us to $\frac{(n-1)!}{x(x+1)\cdots (x+n-1)}$.)
Thus our original sum is equivalent to
$$\int_0^1 \left(\frac{1}{x} - \frac{(n-1)!}{x(x+1)\cdots (x+n-1)}\right) dx = \int_0^1 \left(\frac{1}{x} - \frac{\Gamma(n) \Gamma(x)}{\Gamma(n+x)}\right) dx,$$
which is the formula Andrew mentions in the comments.
Part 2: Rewriting the expression.
Now, rewrite like so:
$$\int_0^1 \left( \frac{1}{x} - \frac{\Gamma(n) \Gamma(x)}{\Gamma(n+x)} \right) dx = \int_0^1 \left( \frac{1}{x}\left(1- \frac{\Gamma(n) \Gamma(x+1)}{\Gamma(n+x)} \right)\right) dx.$$
For $0 < x < 1$, $$\Gamma(x+1) = 1 - \gamma x + \frac{\zeta(2) + \gamma^2}{2}x^2 + O(x^3).$$ (This is the Maclaurin series for $\Gamma(x+1)$; see Expression 8.321 in Gradshteyn and Ryzhik. The only reason I know this is because I had to track it down for a paper I wrote a couple of years ago.) Also, $$\frac{\Gamma(n)}{\Gamma(n+x)} = n^{-x}\left(1 + O\left(\frac{1}{n}\right)\right).$$ (See the DLMF.)
Putting all this together means we want the asymptotic value of
\begin{equation}
\int_0^1 \frac{1-n^{-x}\left(1+ O\left(\frac{1}{n}\right)\right)\left(1 - \gamma x + \frac{\zeta(2) + \gamma^2}{2}x^2 + O(x^3)\right)}{x} dx. \tag{1}
\end{equation}
Part 3: Obtaining the dominant terms.
Following David Speyer and J.M., we'll first extract what turns out to be the dominant part of (1): $$\int_0^1 \frac{1-n^{-x}}{x}dx = \int_0^1 \frac{1-e^{-x \log n}}{x}dx = \int_0^{\log n} \frac{1-e^{-u}}{u} du = \text{Ein}(\log n),$$
where $\text{Ein}(x)$ is the complementary exponential integral. Now, $\text{Ein}(x) = E_1(x) + \log x + \gamma$, where $E_1(x)$ is the usual exponential integral (again, see the DLMF), and $E_1(x) < e^{-x} \log (1 + 1/x)$ (DLMF once again), so putting all of this together we have
$$\int_0^1 \frac{(1-n^{-x})}{x}dx = \log \log n + \gamma + O\left(\frac{1}{n}\right).$$
Part 4: Obtaining the remaining terms.
Now we consider the rest of (1). This is
$$\left(1+ O\left(\frac{1}{n}\right)\right)\int_0^1 n^{-x}\left(\gamma - \frac{\zeta(2) + \gamma^2}{2}x + O(x^2)\right) dx$$
$$=\left(1+ O\left(\frac{1}{n}\right)\right)\left(\gamma \left(\frac{n-1}{n \log n}\right) - \frac{\zeta(2) + \gamma^2}{2}\left(\frac{n - \log n -1}{n (\log n)^2}\right) + O\left(\frac{1}{(\log n)^3}\right)\right)$$
$$=\frac{\gamma }{\log n} - \frac{\zeta(2) + \gamma^2}{2(\log n)^2} + O\left(\frac{1}{(\log n)^3}\right),$$
which is the rest of the expression requested by the OP.