build a tile server using MySQL as backend

You have a long and rocky road ahead of you and the price for "I absolutely cannot use Postgres" will be several person-weeks of work that you have to invest:

• the standard importer, osm2pgsql, cannot import into MySQL; there has been work in something called "osm2mysql" but that is nowhere near feature-complete. Your most likely bet is using GDAL (ogr2ogr) to load OSM data into MySQL. This will mean that you lose the ability to do incremental updates.
• the standard renderer, Mapnik, cannot load data from MySQL. You will have to use Geoserver or MapServer instead.
• the standard map style, OSM-Carto, is built for Mapnik and cannot be used with Geoserver or MapServer; but there are some simpler styles around on the net that work with these renderers.
• the styles that you will find for Geoserver or MapServer will assume a table structure like the one generated by osm2pgsql, but your import process will generate a different table structure; you will either have to modify the styles to match your table structure, or modify your table structure post-import to look like what comes out of osm2pgsql.
• the standard tile serving and queuing mechanism (mod_tile and renderd) only works with Mapnik; you will have to switch to a different solution that is compatible with the rendering engine you choose. Most likely you'll use the renderer in WMS mode and then use mapproxy, or MapServer's mapcache for this.

All in all, you will have to invest a lot of time in this. Are you sure "not using Postgres" is worth it?

And no, there is no step by step guide. You'll have to figure this out yourself.