A search for the associated production of the Higgs boson with a top quark pair is performed in multilepton final states using 20.3 fb$^{-1}$ of proton-proton collision data recorded by the ATLAS experiment at $\sqrt{s}=8$ TeV at the Large Hadron Collider. Five final states, targeting the decays $H\to WW^*$, $ττ$, and $ZZ^*$, are examined for the presence of the Standard Model (SM) Higgs boson: two same-charge light leptons ($e$ or $μ$) without a hadronically decaying $τ$ lepton; three light leptons; two same-charge light leptons with a hadronically decaying $τ$ lepton; four light leptons; and one light lepton and two hadronically decaying $τ$ leptons. No significant excess of events is observed above the background expectation. The best fit for the $t\bar t H$ production cross section, assuming a Higgs boson mass of 125 GeV, is $2.1 ^{+1.4}_{-1.2}$ times the SM expectation, and the observed (expected) upper limit at the 95% confidence level is 4.7 (2.4) times the SM rate. The $p$-value for compatibility with the background-only hypothesis is $1.8σ$; the expectation in the presence of a Standard Model signal is $0.9σ$.