Fix what you know is wrong (cassette and chain) before you go farther with troubleshooting. While you have things apart check the condition of your cables and housings and be sure that the housings are properly prepared and installed; ends cut cleanly and filed flat, and the proper ferrules installed. Pay particular attention to the lower loop of housing going into your derailleur; it tends to collect dirt and crud. Dirty or poorly-installed cables and housings are a frequent cause of shifting woes.