We introduce a Hubble Space Telescope (HST)/Space Telescope Imaging Spectrograph (STIS) stellar census of R136a, the central ionizing star cluster of 30 Doradus. We present low resolution far-ultraviolet STIS spectroscopy of R136 using 17 contiguous 52 arcsec × 0.2 arcsec slits which together provide complete coverage of the central 0.85 parsec (3.4 arcsec). We provide spectral types of 90 per cent of the 57 sources brighter than mF555W = 16.0 mag within a radius of 0.5 parsec of R136a1, plus 8 additional nearby sources including R136b (O4 If/WN8). We measure wind velocities for 52 early-type stars from C IVλλ1548-51, including 16 O2-3 stars. For the first time, we spectroscopically classify all Weigelt and Baier members of R136a, which comprise three WN5 stars (a1-a3), two O supergiants (a5-a6) and three early O dwarfs (a4, a7, a8). A complete Hertzsprung-Russell diagram for the most massive O stars in R136 is provided, from which we obtain a cluster age of 1.5^{+0.3}_{-0.7} Myr. In addition, we discuss the integrated ultraviolet spectrum of R136, and highlight the central role played by the most luminous stars in producing the prominent He II λ1640 emission line. This emission is totally dominated by very massive stars with initial masses above ˜100 M⊙. The presence of strong He II λ1640 emission in the integrated light of very young star clusters (e.g. A1 in NGC 3125) favours an initial mass function extending well beyond a conventional upper limit of 100 M⊙. We include montages of ultraviolet spectroscopy for Large Magellanic Cloud O stars in the appendix. Future studies in this series will focus on optical STIS medium resolution observations.