The Compact Linear Collider (CLIC) is an option for a future e(+) e(-) collider operating at centre-of-mass energies up to 3 TeV, providing sensitivity to a wide range of new physics phenomena and precision physics measurements at the energy frontier. This paper is the first comprehensive presentation of the Higgs physics reach of CLIC operating at three energy stages: root s = 350 GeV, 1.4 and 3 TeV. The initial stage of operation allows the study of Higgs boson production in Higgsstrahlung (e(+) e(-) -> ZH) and WW-fusion (e(+) e(-) -> H nu(e) (nu) over bar (e)), resulting in precise measurements of the production cross sections, the Higgs total decay width Gamma(H), and model-independent determinations of the Higgs couplings. Operation at root s > 1 TeV provides high-statistics samples of Higgs bosons produced through WW-fusion, enabling tight constraints on the Higgs boson couplings. Studies of the rarer processes e(+) e(-) -> t (t) over barH and e(+) e(-) -> HH nu(e) (nu) over bar (e) allow measurements of the top Yukawa coupling and the Higgs boson self-coupling. This paper presents detailed studies of the precision achievable with Higgs measurements at CLIC and describes the interpretation of these measurements in a global fit.