A search for the Standard Model Higgs boson produced in association with a pair of top quarks, where the Higgs boson decays to a pair of bottom quarks, is presented. The search uses 36.1 fb^-1 of p-p collision data at a centre-of-mass energy of √s = 13 TeV collected using the ATLAS detector at the Large Hadron Collider in 2015 and 2016. Focus is placed on the search for the final state where decays of both top quarks produce charged leptons, referred to as the dileptonic final state. The development and study of the statistical fit used to measure the presence of collision events containing Higgs bosons is presented. Results from the combination of the search for the dileptonic final state with the search for the related single-lepton final state are presented. This combined search measured a signal cross-section of µ = 0.84^+0.64_-0.61 times the Standard Model expectation. This result was later combined with searches for final states containing other Higgs boson decays to produce an observation of the production of the Standard Model Higgs boson in association with a pair of top quarks. Developments to the portion of the ATLAS trigger system responsible for track reconstruction are also presented.