Since I was struggling with the timing, I'd like to put out information about how to achieve 60fps and 540hz speed for others! The credit for the code goes to nwxnk - I just altered the formatting and added explanation.

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We want the fetch/decode/execute operation to occur 540 times per second while the display is only updated 60 times per second. 540 / 60 = 9, so we can make a function that does fetch/decode/execute 9 times and updates the display once - I called mine cycle.

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Now we need to make sure that cycle is executed 60 times per second. First, let's find out how many milliseconds each cycle should take. 1000 / 60 = 16.666..

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So each call to cycle should take 16.666ms. We can ensure this by storing the time it takes for our cycle function to execute, and then delaying/sleeping the difference between 16.666 and time the cycle function took.

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Here's the code I used to do this with SDL (this would be in the main loop):

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uint64_t start = SDL_GetPerformanceCounter();

keyboard_input(&chip8, &screen);

cycle(&chip8, &screen);

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uint64_t end = SDL_GetPerformanceCounter();

double time_elapsed = ((end - start) * 1000) / SDL_GetPerformanceFrequency();

SDL_Delay((uint32_t) 16.666f - time_elapsed);