Definition of the Lie coalgebra

Here is one way of defining a Lie coalgebra structure on a vector space $V$. The cobracket is a map $\Theta\colon V\to V\otimes V$ satisfying antisymmetry and co-Jacobi. Antisymmetry means that $\Theta$ induces a map $V\to V\wedge V$. Then co-Jacobi means that if $\Theta(v)=\sum_{i\in I} r_i v_i\wedge w_i$ where $r_i\in\mathbb R$, then $\sum_{i\in I} r_i \Theta(v_i)\wedge w_i + v_i\wedge \Theta(w_i)=0$. One way to think of this is that we have extended $\Theta$ to the entire exterior algebra $\Lambda V$ as a derivation, and the co-Jacobi identity is equivalent to saying that $\Theta^2=0$.

This is dual to one formulation of the Lie algebra axioms. Given a map $b:V\wedge V\to V$, extend it to $\Lambda V$ as a coderivation. Then the Jacobi identity is equivalent to $b^2=0$.

The connection between a Lie algebra $\mathfrak g$ and its dual $\mathfrak g^*$ is that the bracket $b\colon \mathfrak g\otimes \mathfrak g\to \mathfrak g$ is dual to $\Theta\colon \mathfrak g^*\to\mathfrak g^*\otimes \mathfrak g^*$. For example, take $\mathfrak{sl}_2$, with three generators, $E,F,H$ with bracket defined by $$[H,E]=2E,\,\,\, [H,F]=-2F,\,\,\, [E,F]=H.$$

Now take a basis for $\mathfrak{sl}_2^*$ to be $E^*,F^*,H^*$, the dual generators to the basis $E,F,H$. Then we have $$\Theta(H^*)=E^*\otimes F^*-F^*\otimes E^*$$ $$\Theta(F^*)=-\frac{1}{2}(H^*\otimes F^*-F^*\otimes H^*)$$ $$\Theta(E^*)=\frac{1}{2}(H^*\otimes E^*-E^*\otimes H^*)$$ The general rule here is that $\Theta (X)=\sum_{i,j} r_{i,j} X_i\otimes X_j$ where $X_i,X_j$ runs over all pairs of basis elements where $[X_i,X_j]=cX+\cdots$ with $c\neq 0$ and in that case $r_{i,j}:=c^{-1}$.

As far as I can tell, the notion of a Lie coalgebra was first introduced in the paper "Lie coalgebras" by Walter Michaelis (Advances in Mathematics, Volume 38, Issue 1). Although I haven't compared them carefully, I think the definition there matches the one Grumpy Parsnip gave.