Define inductively a sequence of blowups $\varepsilon_n:S_n\ra S_{n-1}$, where $\varepsilon_n:S_n\ra S_{n-1}$ is the blowup of $S_{n-1}$ at a point $p_{n-1}$ and $S_0:=\Bbb P_k^2$. Let $E_n$ be the exceptional divisor of $\varepsilon_n$ on $S_n$ and let $K_n$ be the canonical divisor on $S_n$. Recall that for any divisors $D,D’$ on $S_{n-1}$, we have
$$\varepsilon_n^*D.\varepsilon_n^*D’=D.D’$$
$$\varepsilon_n^*D.E_n=0,\text{ and }$$
$$K_n=\varepsilon_n^*K_{n-1}+E.$$
The plan is to use the Nakai-Moishizon criterion to prove that a necessary condition for $-K_n$ to be ample is $n\le 8$, and if the blowup centers are chosen appropriately, then $-K_n$ is ample iff $n\le 8$. Since $K_{\Bbb P_k^N}\cong \OO_{\Bbb P_k^N}(-N-1)$, we see that $(-K_{\Bbb P_k^2})^2=9$. Moreover,
$$(-K_n)^2=(\varepsilon_n^*K_{n-1}+E)^2=(-K_{n-1})^2-1.$$
By induction,
$$(-K_n)^2=9-n>0\iff n\le 8.$$
Also,
$$(-K_n).E=1>0\quad\forall n\ge1.$$
Lastly, we must check the intersection of $-K_n$ with other curves on $S_n$. We will use that
$$\op{Pic}(\PP{k}{2})\bigoplus_{i=1}^n\Z [E_i] \cong \op{Pic}(S_n).$$
We want to see what minimal restrictions we can place on the centers of the blowups to ensure that $S_n$ is Fano as long as $n\le 8$. If $\tilde C\ne E_n$ is an irreducible curve on $S_n$, we see that either $\tilde C=E_i$ for some $i<n$ or $\tilde C=d\tilde H$ where $\tilde H$ is the composition of strict transforms of a hyperplane. Denote by $m_i$ the multiplicity of $\varepsilon_i\cdot…\cdot\varepsilon_n(\tilde C)$ through $p_{i-1}$. First suppose $\tilde C=E_j$ for some $j$, in which case we have
$(-K_n).\tilde C=-(\varepsilon_n^*K_{n-1}+E).(\varepsilon_n^*\varepsilon_n(\tilde C)-m_nE_n)$
$=(-K_{n-1}).\varepsilon_n(\tilde C)-m_n=…=(-K_j).(\varepsilon_{j+1}\cdot…\cdot\varepsilon_n(\tilde C))-\sum_{i=j+1}^nm_i$
$=1-\sum_{i=j+1}^nm_i.$
This makes sense when you draw a diagram. In general, say you have a smooth projective surface that already has an exceptional divisor, and you blowup up again at a point on that exceptional divisor (an “infinitely near point”). Then, the new exceptional divisor meets the strict transform of the old one transversely. One can check this locally. In other words, here $m_i\in\{0,1\}$ and $m_i=0$ is equivalent to $p_{i-1}\notin E_{i-1}$. Thus, we must choose our blowup centers as to not lie on any previous exceptional curve. So, from now on we assume $S_n$ is the blowup of $\PP{k}{2}$ at $n$ distinct points, none of which are infinitely near any others (none of the blowups “affect” each other). Suppose $\tilde C=d\tilde H$. Then identifying $p_i\in \PP{k}{2}$, $m_i$ is just the multiplicity of $C$ at $p_{i-1}$. Slightly adapting the work above gives
$$(-K_n).\tilde C=3d-\sum_{i=1}^nm_i.$$
Its clear that for any point $p\in\Bbb P_k^2$, we have $\deg C\ge \op{mult}_pC$. We already have a strict bound of $n\le 8$, so we only look at the cases $d=1,2$. If $C$ is a hyperplane, then every point on $C$ has multiplicity 1: $(-K_n).C>0$ if and only if no 3 blowup centers are co-linear. If $C$ is a conic, then points on $C$ can still only have multiplicity 1, even if $C$ is singular (see the remark below). So $(-K_n).C>0$ if and only if no 6 blowup centers lie on a conic. We conclude by Nakai-Moishizon that a sufficient condition for $S_n$ to be Fano is that $n\le 8$ and all blowup centers are in $\PP{k}{2}$ with no 3 co-linear and no 6 lying on the same conic.
Remark: If $S$ is any smooth projective surface, $\varepsilon:\tilde S \ra S$ a blowup, $\tilde C$ the strict transform of a curve $C$ on $S$ that passes through the blowup center with multiplicity $m$, then the (arithmetic) genus formula gives,
$$g(\tilde C)=\frac{(\varepsilon^*C-mE).(\varepsilon^*C-mE+\varepsilon^*K_S+E)}{2}+1=g(C)-\frac{m(m-1)}{2}.$$
In particular, if $g(C)=0$, as in the case of degree 1 and 2 projective plane curves, then $m\in\{0,1\}$.
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