  This paper initiates a systematic study of the cyclotomic KLR algebras of
affine types $A$ and $C$. We start by introducing a graded deformation of these
algebras and the constructing all of the irreducible representations of the
deformed cyclotomic KLR algebras using content systems and a generalisation of
the Young's seminormal forms for the symmetric groups. Quite amazingly, this
theory simultaneously captures the representation theory of the cyclotomic KLR
algebras of types $A$ and $C$, with the main difference being the definition of
residue sequences of tableaux. We then use our semisimple deformations to
construct two "dual" cellular bases for the non-semisimple KLR algebras of
affine types $A$ and $C$. As applications of this theory we recover many of the
main features from the representation theory in type $A$, simultaneously
proving them for the cyclotomic KLR algebras of types $A$ and $C$. These
results are completely new in type $C$ and we, usually, more direct proofs in
type $A$. In particular, we show that these algebras categorify the irreducible
integrable highest weight modules of the corresponding Kac-Moody algebras, we
construct and classify their simple modules, we investigate links with
canonical bases and we generalise Kleshchev's modular branching rules to these
algebras.
