LISA is a future space-based gravitational wave detector that will complement the LIGO/Virgo observations at much lower frequencies, enabling the detection (among other targets) of coalescences of massive black hole binaries, and of early inspirals of stellar-mass black holes far from merger. The development of data analysis tools for LISA is still in its exploratory phase, and it is crucial to understand the capabilities of LISA and the trade-offs in its instrumental design. Previous studies often used inspiral-only signals and the Fisher matrix approximation. We present a set of tools that allows fast likelihood computations for Fourier-domain waveform models, enabling Bayesian analyses exploring the full parameter space. We present examples of simulated parameter recovery for massive black hole binaries, highlight degeneracies in parameter space and explain the role of both frequency-dependent effects in the instrument response and of higher harmonics in the signal in breaking these degeneracies. We also present results for the parameter extraction of stellar-mass black holes.