Let
Q
Q
be a finite quiver without oriented cycles, and let
Λ
\Lambda
be the corresponding preprojective algebra. Let
g
\mathfrak {g}
be the Kac-Moody Lie algebra with Cartan datum given by
Q
Q
, and let
W
W
be its Weyl group. With
w
∈
W
w \in W
, there is associated a unipotent cell
N
w
N^w
of the Kac-Moody group with Lie algebra
g
\mathfrak {g}
. In previous work we proved that the coordinate ring
C
[
N
w
]
\mathbb {C}[N^w]
of
N
w
N^w
is a cluster algebra in a natural way. A central role is played by generating functions
φ
X
\varphi _X
of Euler characteristics of certain varieties of partial composition series of
X
X
, where
X
X
runs through all modules in a Frobenius subcategory
C
w
\mathcal {C}_w
of the category of nilpotent
Λ
\Lambda
-modules. The first aim of this article is to compare the function
φ
X
\varphi _X
with the so-called cluster character of
X
X
, which is defined in terms of the Euler characteristics of quiver Grassmannians. We show that for every
X
X
in
C
w
\mathcal {C}_w
,
φ
X
\varphi _X
coincides, after an appropriate change of variables, with the cluster character of Fu and Keller associated with
X
X
using any cluster-tilting object
T
T
of
C
w
\mathcal {C}_w
. A crucial ingredient of the proof is the construction of an isomorphism between varieties of partial composition series of
X
X
and certain quiver Grassmannians. This isomorphism is obtained in a very general setup and should be of interest in itself. Another important tool of the proof is a representation-theoretic version of the Chamber Ansatz of Berenstein, Fomin and Zelevinsky, adapted to Kac-Moody groups. As an application, we get a new description of a generic basis of the cluster algebra
A
(
Γ
_
T
)
\mathcal {A}(\underline {\Gamma }_T)
obtained from
C
[
N
w
]
\mathcal {C}[N^w]
via specialization of coefficients to 1. Here generic refers to the representation varieties of a quiver potential arising from the cluster-tilting module
T
T
. For the special case of coefficient-free acyclic cluster algebras this proves a conjecture by Dupont.