Characterize the free $\ell$-group on one generator where an $\ell$-group is an algebra of the form (G,∧,∨,·,$^{−1}$,1), where (G,·,$^{−1}$,1) is a group, (G,∧,∨) is a (distributive) lattice and multiplication is order-preserving and distributes over join and meet.
I was thinking that if $x$ is the generator then we would have a lattice with an antichain
$$... (x^{-1})^{2}\hspace{3mm} x^{-1}\hspace{3mm} 1\hspace{3mm} x \hspace{3mm}x^2...$$
with the joins and meets above and below which satisfy the $\ell$-group conditions, i.e. $x^2 \vee 1 = x \cdot(x \vee x^{-1})$ but I'm not sure what else I can say about it. It seems like the only other thing I could use is the universal property but I'm not sure how that would help.
Yes, you have the antichain you mention. Now generate a free distributive lattice with that antichain. The result is the free $\ell$-group on one generator. When done, you get that the free $\ell$-group on one generator is isomorphic to $\mathbb Z\times \mathbb Z$ with coordinatewise operations and order. It is freely generated by $x=(1,-1)$.
This fact is the content of Theorem 17 of
Birkhoff, Garrett
Lattice-ordered groups. Ann. of Math. (2) 43 (1942), 298-331.