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The first step is cleaning the mattress is removing the vomit from the surface of the bed. Use a paper plate to scrape any solids off the bed, and place it in a plastic bag to discard in the trash. Before cleaning up the vomit, it's a good idea to put on rubber gloves. That will protect you from any germs. You can also use a dustpan to scrape the vomit from your bed and dump it in the toilet to dispose of it. Hose the dustpan down outside to clean it. If the bedding is still on your bed, remove them before moving on to cleaning the mattress. Strip the sheets, comforter, mattress pad, and any other items from the bed, and wash them in the washing machine. Wash the bedding on the highest temperature setting that is available on the washer. That will help kill any lingering germs. Once you've removed the sheets from the bed, use a dry cloth to soak up any liquid from the vomit that may have reached the mattress. Avoid rubbing the stained area, though. Instead, blot it to help remove the liquid without spreading it around. It's a good idea to use an old rag to blot the mattress because you may want to throw it out when you're done.
Scrape the vomit off the bed. Remove the sheets and wash thoroughly. Soak up the remaining liquid from the mattress.