A measurement of the associated production of a top-quark pair with the Higgs boson (tt¯H) in multilepton final states is presented. The analysis is based on a data sample of proton-proton collisions at s=13 TeV recorded with the ATLAS detector at the CERN Large Hadron Collider and corresponding to an integrated luminosity of 140 fb−1. Six final states defined by the number and flavour of reconstructed charged leptons are combined in a simultaneous likelihood fit to extract the tt¯H signal and constrain the most relevant backgrounds. The measured tt¯H cross-section normalised to Standard Model (SM) prediction is σtt¯H/σSM=0.63−0.19+0.20. This result corresponds to an observed (expected) significance of 3.3σ (5.3σ). Additionally, two other fits are used to measure the tt¯H cross-section differentially in bins of the Higgs boson transverse momentum in the simplified template cross-section framework, and to extract the associated production cross-section of a single top-quark with the Higgs boson (tH) together with the tt¯H one. The CP structure of the top quark-Higgs boson Yukawa coupling is probed through an analysis of tt¯H and tH events. The results are compatible with the SM hypothesis, and values of the mixing angle between CP-even and CP-odd top-Higgs Yukawa couplings of |α| > 62° are excluded at 68% confidence level.