I believe the term chain was introduced by Poincare as part of his second definition of the term manifold; the definition I am referring to is on pg. 24-25 of [Stillwell's wonderful translation of Analysis Situs](http://www.maths.ed.ac.uk/~aar/papers/poincare2009.pdf). 

> "We then say that the two manifolds V and V ′ are analytic
> continuations of each other. In this way we can form a chain of
> manifolds V1, V2, ..., Vn such that each is an analytic continuation
> of its predecessor, and there is a common part between any two
> consecutive manifolds of the chain. I shall call this a connected
> chain."

Poincare goes on and mentions something akin to a combination of oriented simplices: 

> "...There are, in fact, manifolds (and we shall see examples later)
> which can be decomposed into a certain number of partial manifolds
> forming a connected chain or network and such that each of them can be
> defined by equations of the form (8)..." 

[(8) references that a collection of equations of the form
$x_1= \theta_1(y_1,...y_m) ; x_2=\theta_2(y_1,...y_m) ;...; x_n=\theta_n(y_1,...y_m) $ represents a manifold of m dimensions (if the $y$ are considered as independent variables).]

Also note, Alexander's definition of the term chain (page 301 of Combinatorial Analysis Situs) cites Poincare's formula for describing the cell structure of a manifold in terms of an incidence matrix.

[![enter image description here][1]][1]

 

> "In his first long paper on topology Poincare (1895) continued to
> explore the Betti numbers. He set up machinery for computing them by
> assuming that each manifold has a decomposition into cells
> homeomorphic to simplices, reading off linear equations he called
> homologies, and computing the Betti numbers by linear algebra."

-- [Mathematical perspectives](http://www.ams.org/journals/bull/2012-49-04/S0273-0979-2012-01385-X/S0273-0979-2012-01385-X.pdf), Stillwell

  [1]: https://i.sstatic.net/iX62j.png