(*) Jupiter, for all intents and purposes, doesn't have a solid surface to stand on. Not any more than you could say that Earth's atmosphere has it, before you hit Terra Firma. It's an enormous ball composed of mostly Hydrogen and Helium, but also other heavier elements in smaller parts, and it's so massive that its own gravity compresses these gases into liquid the further into its interior we go. Lighter elements dominate in its upper atmosphere in gas state; these gradually compress due to its own pressure into liquids, deeper still metallic Hydrogen, and eventually mesh of metallic Hydrogen, rock, and other heavier elements that sunk deeper into its core. Nobody would be able to "stand" on any of these layers. In fact the temperature and pressure become so great, it's been calculated that even diamonds (it's speculated they could form as precipitates in certain layers of Jupiter's interior from black clouds of soot, where, if proven true, would mean it quite literally "rains diamonds") eventually melt into, again speculated, gooey form of liquid carbon that's not so much unlike tar, except that it isn't.
A pie slice of Jupiter's pressure, temperature and density layers. Source:
Laboratory for Atmospheric and Space Physics, University of Colorado at Boulder
So what would happen if you dived into Jupiter? Well, how long you'd last would depend on what gear you're in that protects you from its hostile environment. Pressure would at first gradually increase in its upper atmosphere to the point that it's sufficient for its violent storms to throw you around. That's the Jupiter's cloud layer. You might be "lucky" though, and fall into it at its poles where giant atmospheric depressions keep all of it somewhat lower, slightly prolonging the inevitable outcome. As the pressure increases, so does thermal convection. You'd start losing heat increasingly more rapidly, and it's not a nice Mediterranean spring temperature either. At one atmosphere (pressure equal to mean sea-level on Earth) the temperature goes as low as −108 °C. That's below the coldest temperatures ever recorded on Earth's surface (~ −93 °C in the East Antarctic Plateau), even for its polar regions during winters. All the while, you'd also be bombarded by Jupiter's radiation. And if you're falling into it from its poles and you thought you were lucky for a few hundred kilometers more, think again because those are the regions where Jupiter magnetically reconnects with Sun's own magnetic field, increasing speed of charged particles to the point that we can observe fantastic "electric blue" polar aurorae the size of many Earths where this Solar proton flux ionizes Jupiter's upper atmosphere.
So you have three main adversaries to fight against with your environmental protective gear you're in: radiation, pressure, and temperature. And if you enter its upper atmosphere too fast, also contact ionisation, triboelectric charge, surface ablation… nothing too charming and all of it quite terminating on its own. When would any of these be too much to stand and your equipment fails is anyone's guess, but it wouldn't take very long at Jupiter's gravity (24.79 m/s²), regardless of your initial descent rate, until you dive too deep for comfort.
Eventually, once long dead from gas giant inhospitability, your remains would submerge deeper into Jupiter's liquid Hydrogen layer. First freeze solid, then thaw as the temperature and pressure increase to nearly 5,000 °C and about 2 million times Earth's sea-level atmospheric pressure. You'd nearly implode, if your body wasn't mostly water, which doesn't compress easily. You would still compress greatly as all your body's once functioning cavities collapse. Not the best time for a selfie. Your journey isn't yet over though, because you and your gear you're in are still denser than that particular Jovian layer and would sink deeper still towards its metallic Hydrogen layer that starts at a density of roughly 1 g/cm3 and continues to nearly 25 g/cm3 (with average density of ~ 4 g/cm3, or slightly more than 4 times your own body's density, if we excluded an EVA suit you'd have to be in, adding to your overall density. At that point, you're being zapped by tremendous electric currents that give Jupiter such an enormous magnetosphere, the second largest structure in our Solar system besides the Sun's own heliosphere.
These currents would tear your remains apart into indistinguishably small fragments, and induce chemical decomposition by free atomic Hydrogen radicals randomly exchanging electrons. It would look a bit like submerging a body into hydrofluoric acid while deep frying it at the same time, if perhaps not more violent. I don't know, I can only imagine, I've never actually done it. Honest! Anyway, fragments of what was once you would decompose into its constituent chemical elements, lose valence and bound with surrounding free hydrogen protons. Heavier compounds would sink even deeper, where the pressure and current would eventually cause them to lose hydrogen protons and recombine with themselves or other heavier elements and electron hungry molecules present into so compressed and hot state not even current science is able to tell their exact nature and behavior.
Either case, you'd be spread all over Jupiter's interior in various states, and become a part of it for near to eternity. Pretty epic, but please don't do it.
(*) Not all of it might be necessarily exactly true, since some parts I describe are a subject to still ongoing research, but this was kinda fun so I went for it. I'll revise to add some references at a later time for the parts that there are any available.