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We give an overview and describe the rationale, methods, and testing of the Hubble Space Telescope (HST) Archival Legacy project "SKYSURF." SKYSURF uses HST's unique capability as an absolute photometer to measure the ~0.2-1.7 $\mu$m sky surface brightness (SB) from 249,861 WFPC2, ACS, and WFC3 exposures in ~1400 independent HST fields. SKYSURF's panchromatic dataset is designed to constrain the discrete and diffuse UV to near-IR sky components: Zodiacal Light (ZL; inner Solar System), Kuiper Belt Objects (KBOs; outer Solar System), Diffuse Galactic Light (DGL), and the discrete plus diffuse Extragalactic Background Light (EBL). We outline SKYSURF's methods to: (1) measure sky-SB levels between its detected objects; (2) measure the integrated discrete EBL, most of which comes from AB$\simeq$17-22 mag galaxies; and (3) estimate how much diffuse light may exist in addition to the extrapolated discrete galaxy counts. Simulations of HST WFC3/IR images with known sky-values and gradients, realistic cosmic ray (CR) distributions, and star plus galaxy counts were processed with nine different algorithms to measure the "Lowest Estimated Sky-SB" (LES) in each image between the discrete objects. The best algorithms recover the inserted LES values within 0.2% when there are no image gradients, and within 0.2-0.4% when there are 5-10% gradients. SKYSURF requires non-standard re-processing of these HST images that includes restoring the lowest sky-level from each visit into each drizzled image. We provide a proof of concept of our methods from the WFC3/IR F125W images, where any residual diffuse light that HST sees in excess of the Kelsall et al. (1998) Zodiacal model prediction does not depend on the total object flux that each image contains. This enables us to present our first SKYSURF results on diffuse light in Carleton et al. (2022)., Comment: Accepted to AJ; see accompanying paper Carleton et al. 2022: arXiv:2205.06347. Comments welcome!