You can publish a hosted tiled map service from your data by authoring a map in ArcGIS Desktop.
Sign in to arcgis online from desktop (File menu)
Author your map
Choose the File menu > Share as a service
Choose my hosted services and step through the wizard.
You will want to greatly restrict the number of scales you build tiles for, otherwise you'll use up all your free credits.
Another option would be to create a tile package and upload it to arcgis online. There is a Geoprocessing tool to do so. Once uploaded, you can create a service from it. This will save you the credits for building the cache in ArcGIS Online.
You'll probably need to read some documentation, but this gives you a start.
Good luck,
Mike