The original entrance to OFD is right at the bottom of the hill and the bottom of the cave. It is possible to travel uphill in the cave all the way to Smith's Armoury which is the point furthest from any entrance, travelling almost the whole Nant Newydd "Main Streamway", which is split into OFD I, II and III (Roman numerals). Dry routes to bypass the sumps between these sections of the stream are well-trodden and well-described on ogof.org.uk, but it is a very long trip and it's advisable to learn each section first. The "Top" entrance which provides easiest access to the OFD II and OFD III streamways, and the routes between them, is notoriously complex and a survey will serve you better than a description. Smith's Armoury lies not far from where the water sinks on the surface and enters the cave, so it's not really that deep underground, but given the effort, skill and hazard required to get there it definitely feels very far from home.
As Andy says the deepest below the surface is probably in the streamway. I would guess somewhere near the Top Waterfall but there's bound to be some absolute nerd who can tell you exactly.
There are many places that are even more obscure and less frequented, taking a longer time to get to and arguably further from the entrances. Merthyr Vale is apparently rather difficult to get to from anywhere and could also be a contender.