We study magnetism in the κ-(ET)₂X family of charge-transfer salts using implanted muon spectroscopy in conjunction with detailed ab initio electronic structure calculations using density functional theory (DFT). ET stands for the electron donor molecule bis(ethylendithio)tetrathiafulvalene and X is an anion. The DFT calculations are used to establish molecular spin distributions, muon stopping sites, and dipolar field parameters, that allow us to make a quantitative interpretation of the experimental results. Materials in the κ-(ET)₂X family with X = Ag₂(CN)₃ and X = Cu₂(CN)₃ have attracted particular interest, as they have the attributes of quantum spin liquids, showing no magnetic ordering down to 30 mK in zero field μSR and in NMR, despite having exchange couplings of order 200-250 K. In contrast, the material with X = Cu[N(CN)₂]Cl has an antiferromagnetic (AF) ordering transition with T_N in the region of 23–30 K. In order to better understand the muon spectroscopy signature of magnetism in this whole family of compounds at both low and high magnetic fields, we look in detail at the case X = Cu[N(CN)₂]Cl. As the first step in our study, the spin density distribution for the ET dimer is calculated using DFT and used to simulate the 3.7 T ¹H-NMR spectrum of this salt, with the spectrum showing good agreement with that measured previously [K. Miyagawa, A. Kawamoto, Y. Nakazawa, and K. Kanoda, Phys. Rev. Lett. 75, 1174 (1995)]. Best match to the data is found for antiferromagnetic interlayer ordering and an ordered moment per dimer of 0.25 μ_B. DFT is also used to explore muon stopping sites for this salt, finding one set of sites resulting from muonium addition to C=C double bonds in the ET layer, with muons stopping in the anion layer forming another group of sites. The dipolar fields associated with each of the stopping sites is computed and these are compared with the precession frequencies observed in the ZF-μSR spectrum [M. Ito, T. Uehara, H. Taniguchi, K. Satoh, Y. Ishii, and I. Watanabe, J. Phys. Soc. Jpn. 84, 053703 (2015)]. Best match to the ZF-μSR spectrum is obtained with the mode of interlayer ordering having FM character and an ordered moment per dimer of 0.31 μ_B for muon sites in the anion layer and 0.36 μ_B for muonium sites in the ET layer. New measurements of TF-μSR spectra for fields up to 8 T are reported and analyzed to obtain the best estimate of the magnetic order parameter under different measurement conditions, allowing us to observe the variation of T_N with applied field and the field-induced transverse canting of the moments.