How do I clean linoleum flooring thats dirty without painting it?

Hi Shauna, you can try the method in the attached video.

https://www.google.ca/search?q=clean+dirty+linoleum&rlz=1C1GCEA_enCA815CA815&oq=clean+dirty+lin&aqs=chrome.1.69i57j0l5.8065j0j8&sourceid=chrome&ie=UTF-8#kpvalbx=1

Or you can try this method,

Before you clean your linoleum floor, sweep or vacuum to remove excess dirt. Then, make a 1:1 vinegar-water cleaning solution, or add 6-7 drops of dishwashing liquid to a gallon of warm water to make your solution. Use a scrub brush and microfiber cloth. Good luck!

Also, you can use Future floor coating to keep is sealed once you get it cleaned

https://www.amazon.com/Pledge-11182-Revive-Transparent-Liquid/dp/B000ARPH4C

Try using one cup of lemon scented ammonia to one gallon hot water. Removes ground in dirt, grime, and wax build up.

Try using TSP. It's a heavy duty floor cleaner https://www.homedepot.com/p/SAVOGRAN-1-lb-Box-TSP-Heavy-Duty-Cleaner-10621/202935861