There is a brief description of the double cover of an even lattice on page 2 of Borcherds's paper *Vertex algebras, Kac-Moody algebras and the monster* (number 4 on the page). It is introduced there as a set of properties that uniquely define it up to isomorphism, but without a construction. For a detailed construction, see Chapters 5 and 7 of *Vertex Operator Algebras and the Monster* by Frenkel, Lepowky, and Meurman.

The automorphism group of the double cover of an even lattice $L$ is an extension of $\operatorname{Aut}(L)$ by $(\mathbb{Z}/2\mathbb{Z})^{\text{rank}(L)}$, and it is usually non-split. For the Leech lattice, you do not get the monster, but you get something closely related to a large subgroup. In more detail, the automorphism group of the double cover of Leech naturally acts on the Leech lattice vertex algebra, and this vertex algebra in turn can be used to construct the monster vertex algebra using a "twisted module". The automorphism groups then yield a diagram of the following form:
$$2^{24}.Co_0 \to 2^{24}.Co_1 \leftarrow 2^{1+24}.Co_1 \hookrightarrow \text{Monster}.$$
The leftmost group is the automorphism group of the double cover of Leech, the second group is the image of its action on a fixed point subalgebra of the Leech lattice vertex algebra under an involution, the third group is a central extension that acts on a fixed point submodule of the twisted module, and it is the centralizer of an element of order 2 in the monster. Frenkel, Lepowsky, and Meurman constructed the monster action by extracting extra symmetry from the direct sum of the fixed point subalgebra and the fixed point submodule - this required a large fraction of a book.