Surface X-ray diffraction has been used to analyze in situ the room-temperature adsorption behaviour and the structure of K and Cs on Cu(001) at submonolayer coverages. Adsorption of K takes place in fourfold hollow sites up to coverages of about 0.25 monolayers (ML), where 1 ML corresponds to 1.53 x 10(15) atoms/cm2. At higher coverages the formation of a quasi-hexagonal incommensurate adlayer is observed. In contrast, for Cs adsorption we observe from the very beginning the formation of the quasi-hexagonal structure up to the completion of the adlayer at about 0.30 ML. For K adsorption in the hollow sites we determine an adsorption height, d(perpendicular-to) 2.25(15) angstrom, corresponding to an effective K radius of r(eff) = 1.6(1) angstrom close to the ionic radius of 1.33 angstrom. We do not observe a change in the effective radius as a function of coverage. For the quasi-hexagonal Cs structure we find an (average) adsorption height d(perpendicular-to) = 2.94 angstrom corresponding to an effective radius of r(eff) = 2.18 and 1.93 angstrom, for the limiting cases of hollow- and bridge-site adsorption, respectively. The analysis of the superlattice reflections corresponding to the quasi-hexagonal incommensurate structures indicated that the K adlayer is strongly modulated. The first Fourier component of the substrate-induced modulation was determined to u01 = 1.29(3) angstrom. In contrast, for Cs/Cu(001) static modulation is much less important (u01 less-than-or-equal-to 0.2 angstrom). Variation of the Cs adlayer density by changing the substrate temperature allows continuous expansion and contraction of the adsorbate unit cell. No commensurate-incommensurate transition has been observed.