  The strong-coupling character expansion of lattice models is reanalyzed in
the perspective of its complete algorithmization. The geometric problem of
identifying, counting, and grouping together all possible contributions is
disentangled from the group-theoretical problem of weighting them properly. The
first problem is completely solved for all spin models admitting a
character-like expansion and for arbitrary lattice connectivity. The second
problem is reduced to the evaluation of a class of invariant group integrals
defined on simple graphs. Since these integrals only depend on the global
symmetry of the model, results obtained for principal chiral models can be used
without modifications in lattice gauge theories. By applying the techniques and
results obtained we study the two-dimensional principal chiral models on the
square and honeycomb lattice. These models are a prototype field theory sharing
with QCD many properties. Strong-coupling expansions for Green's functions are
derived up to 15th and 20th order respectively. Large-$N$ and $N=\infty$
results are presented explicitly. Related papers are devoted to a discussion of
the results.
