The final ATLAS Run 1 measurements of Higgs boson production and couplings in the decay channel $H\ensuremath{\rightarrow}Z{Z}^{*}\ensuremath{\rightarrow}{\ensuremath{\ell}}^{+}{\ensuremath{\ell}}^{\ensuremath{-}}{\ensuremath{\ell}}^{\ensuremath{'}+}{\ensuremath{\ell}}^{\ensuremath{'}\ensuremath{-}}$, where $\ensuremath{\ell}$, ${\ensuremath{\ell}}^{\ensuremath{'}}=e$ or $\ensuremath{\mu}$, are presented. These measurements were performed using $pp$ collision data corresponding to integrated luminosities of 4.5 and $20.3\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ at center-of-mass energies of 7 and 8 TeV, respectively, recorded with the ATLAS detector at the LHC. The $H\ensuremath{\rightarrow}Z{Z}^{*}\ensuremath{\rightarrow}4\ensuremath{\ell}$ signal is observed with a significance of 8.1 standard deviations, with an expectation of 6.2 standard deviations, at ${m}_{H}=125.36\text{ }\text{ }\mathrm{GeV}$, the combined ATLAS measurement of the Higgs boson mass from the $H\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\gamma}$ and $H\ensuremath{\rightarrow}Z{Z}^{*}\ensuremath{\rightarrow}4\ensuremath{\ell}$ channels. The production rate relative to the Standard Model expectation, the signal strength, is measured in four different production categories in the $H\ensuremath{\rightarrow}Z{Z}^{*}\ensuremath{\rightarrow}4\ensuremath{\ell}$ channel. The measured signal strength, at this mass, and with all categories combined, is ${1.44}_{\ensuremath{-}0.33}^{+0.40}$. The signal strength for Higgs boson production in gluon fusion or in association with $t\overline{t}$ or $b\overline{b}$ pairs is found to be ${1.7}_{\ensuremath{-}0.4}^{+0.5}$, while the signal strength for vector-boson fusion combined with $WH/ZH$ associated production is found to be ${0.3}_{\ensuremath{-}0.9}^{+1.6}$.