What is going on with this constrained optimization? I'd like to figure out what is going on when trying to maximize a function (below $a_i$ are real numbers)
$F = a_1a_2 + a_2a_3 + \cdots + a_{n - 1}a_n  + a_na_1;$
When we have active constraints
$h_1 = a_1 + a_2 + \cdots + a_n = 0;$
$h_2 = a_1^2 + a_2^2 + \cdots +a_n^2 = 1;$

So my gradients
$\nabla F = (a_2+a_n,a_3+a_1, \ldots , a_{n-2}+a_n,a_{n-1}+a_1);$
$\nabla h_1 = (1, 1, \ldots , 1);$
$\nabla h_2 = (2a_1,2a_2,\ldots , 2a_n);$
Kuhn-Tucker should provide necessary conditions in this case which is I guess pretty much the same as the method of Lagrange multipliers:
$$\begin{cases}
a_2+a_n = \lambda_1 + 2\lambda_2a_1; \\
a_3+a_1 = \lambda_1 + 2\lambda_2a_2; \\
a_4+a_2 = \lambda_1 + 2\lambda_2a_3; \\
\ldots \\
a_{n-1} + a_1 = \lambda_1 + 2\lambda_2 a_n;
\end{cases}$$
In a matrix form this is
$$\begin{pmatrix}
2\lambda_2 & -1 & 0 & 0 & \ldots & 0 & -1 \\
-1 & 2\lambda_2 & -1 & 0 & \ldots & 0 & 0 \\
0 & -1 & 2\lambda_2 & -1 & \ldots &0 & 0 \\
\ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\
0 &0 &0 & 0 & \ldots & 2\lambda_2 & -1 \\
-1 &0 &0 & 0 & \ldots & -1 & 2\lambda_2
\end{pmatrix}
\begin{pmatrix}
a_1 \\ a_2 \\ a_3 \\ \vdots \\ a_{n-1} \\ a_n
\end{pmatrix} =
- \lambda_1\begin{pmatrix}
1 \\ 1 \\ 1 \\ \vdots \\ 1 \\ 1
\end{pmatrix}
$$
How should I proceed?

Well, on adding up the n equations you have, you get
$$2\sum_{i=1}^{n}a_i = n\lambda_1 + 2\lambda_2\sum_{i=1}^{n}a_i$$
which gives, $\lambda_1 = 0$ as $\sum_{i=1}^{n}a_i=0$.
So now you need to find the nullspace of the matrix, which should hopefully be easier?
