This paper investigates channel coding for DNA data storage, a recent and emerging paradigm in which both substitution errors and synchronization errors (insertions, deletions) are introduced in the stored data. The paper first proposes a novel and accurate statistical channel model to represent errors introduced by the DNA data storage support. The channel model takes the form of a k-order Markov Model which allows to capture both memory in the sequence of channel errors, and statistical dependency between errors and input sequence. The model probabilities are inferred on two sets of data: one set of experimental data, and one set of genomic data. In both cases, we observe a superior accuracy, evaluated from the Kulback-Leibler divergence, of our channel model compared to existing ones. Second, the paper investigates the design of error-correction solutions dedicated to the proposed channel model. It considers a concatenated code construction built from a convolutional decoder which aims to correct synchronization errors, and from an outer LDPC code which corrects residual substitution errors. While this construction was already investigated for i.i.d. errors, this paper shows how to include all the knowledge of the proposed channel model into the convolutional decoder. Numerical results show that the proposed decoding algorithm significantly improves the decoding performance in terms of BER and FER, at the price of an increased decoding complexity.