The production cross-section of J/psi pairs in proton-proton collisions at a centre-of-mass energy of root s = 13TeV is measured using a data sample corresponding to an integrated luminosity of 4.2 fb(-1) collected by the LHCb experiment. The measurement is performed with both J/psi mesons in the transverse momentum range 0 < p(T) < 14 GeV/c and rapidity range 2.0 < y < 4.5. The cross-section of this process is measured to be 16.36 +/- 0.28 (stat) +/- 0.88 (syst) nb. The contributions from single-parton scattering and double-parton scattering are separated based on the dependence of the cross-section on the absolute rapidity difference Delta y between the two J/psi mesons. The effective cross-section of double-parton scattering is measured to be sigma(eff) = 13.1 +/- 1.8 (stat) +/- 2.3 (syst) mb. The distribution of the azimuthal angle phi(CS) of one of the J/psi mesons in the Collins-Soper frame and the p(T)-spectrum of the J/psi pairs are also measured for the study of the gluon transverse-momentum dependent distributions inside protons. The extracted values of < cos4 phi(CS)> and < cos2 phi(CS)> are consistent with zero, but the presence of azimuthal asymmetry at a few percent level is allowed.