Pretty easy and straight forward to do. Simply go to the Time Trial option, set up your session. Select any options as you please. Drive round the track, ensuring you don't cut corners or crash into walls. Slow and steady for 1 lap to make sure it is a clean lap, then you can push after the 1st clean lap, if you wish to attempt a good pace.
Rinse and repeat for 10 different circuits. I like to write down the circuits I have completed in my phone notes app, so I don't waste time going into a circuit, to find out I have already completed it.
You can aim to complete this achievement at the same time if you wish. Both achievements stack on top of one another - so you can complete 5 different tracks in the wet for the below achievement, then do 5 different tracks in either dry or wet conditions to complete both achievements in 10 different circuits.