How cheap can I go on a PC for Pokémon and N64 ROM hacks?

You can spend well under $1,000. A used office desktop with an Intel Core i5 from the 8th or 9th generation is a reasonable place to start. You don't need a dedicated graphics card just to get started with Pokémon and N64 emulation.

Check the condition and what comes with it before buying. Small office PCs can be awkward to upgrade, so if you think you might want a graphics card later, check the case size and power supply first.

A new mini PC is another option if you don't want to hunt for a used machine. Look for a modern processor with integrated graphics and 16GB of RAM. It's a compact setup and should be plenty for typical N64 emulation without paying for a gaming tower.

Fangames aren't all the same, though. Check the operating system and requirements for any specific fangame you care about before buying. An emulator running well doesn't automatically mean every standalone game will.

If you want to build a desktop, go with a processor that has integrated graphics and skip the separate GPU for now. Make sure the particular CPU you pick actually includes graphics, then choose a compatible motherboard and RAM. That leaves you room to add a GPU later if your interests change.

Does it need to be a desktop? A retro handheld could be a much cheaper way to play on its own screen. A PC handheld is another option and can be connected to a larger display with a suitable dock.

Check compatibility for the specific hacks and fangames you want before choosing one. A device that runs the original console's games may not run every hack equally well, and a standalone fangame may need a particular operating system.

If you want a little room to try more demanding emulators later, a used desktop with a GTX 1650 or 1660 is worth considering. That's more graphics hardware than your stated N64 and Pokémon goal calls for, so I wouldn't pay extra for it unless that flexibility appeals to you.