  A Pfaffian-type Murnaghan-Nakayama rule is derived for the Hecke-Clifford
algebra $\mathcal{H}^c_n$ based on the Frobenius formula and vertex operators,
and this leads to a combinatorial version via the tableaux realization of
Schur's $Q$-functions. As a consequence, a general formula for the irreducible
characters $\zeta^{\la}_{\mu}(q)$ using partition-valued functions is derived.
Meanwhile, an iterative formula on the indexing partition $\la$ via the Pieri
rule is also deduced. As applications, some compact formulae of the irreducible
characters are given for special partitions and a symmetric property of the
irreducible character is found. We also introduce the spin bitrace as the
analogue of the bitrace for the Hecke algebra and derive its general
combinatorial formula. Tables of irreducible characters are listed for
$n\leq7.$
