To complete Tonic's challenge you must first free him from the Gungeon. Whilst playing naturally you will come across gold keys which means when you find a gold key, there will be a jail cell with a random npc on the same level. Simply unlock the cell with the gold key and speak to the npc and he/she will move to the breach. Tonic is a big blue dog who once freed, be located in the same room as Daisuke in the breach (room to the right of the Gungeon door).
To activate the challenge, speak to tonic and he will toggle turbo mode. Turbo mode increases the overall game speed of both the enemies and yourself. You have to start from chamber one for the achievement and it will unlock as soon as you kill the dragon in chamber 5.
I recommend to only attempt this when you can comfortably run through the game and know all of the enemies and bosses attack patterns as turbo speed can be quite challenging but this achievement also relies on getting good RNG for guns and passive abilities.