Just been through the same issue but had some drainage issues on top. Ours was sloped, dips everywhere and we needed to remove a path and replace it with grass. Read through hundreds of forums and pretty sure half of them are making it up.
Ended up getting a rotavator from Screwfix for £50 and turned the lawn over mixing in sharp sand from B&Q (https://www.diy.com/departments/tarmac-sharp-sand-large-bag/5030349012445_BQ.prd) - helped break up the clay. Leveled out with a rake and then topsoil, sand and seed mix, water loads, and then another layer of topsoil.
If you get some word you can make a DIY leveler and just drag it around. Saves the rake.