The best way I have found to compost lawn cuttings is to place them in a wooden compost bin, alternating layers between grass and soil. Depending on how much grass you have you may need a couple of bins in use. When I had a small garden I used a plastic dustbin, with holes drilled in the bottom.
It doesn’t matter what you use, the process is the same, a layer of soil in the bottom, then a 150mm-250mm layer of grass cuttings, then another layer of soil. This soil layer only has to cover the grass, 50mm is enough, and then another layer of grass.
The soil introduces lots of bacteria into the compost bin, and acts as a sponge to soak up the liquid in the grass. If grass is left in a thicker layer it doesn’t break down because it shuts out any air and can get into a wet sticky mess, which is then very difficult to do anything with.
The soil also helps squash down the cuttings as they are very bulky, which gives more space for the next addition to the bin. The grass also heats up the contents which further accelerates the decomposition process.
The best compost bins I have used have been made of wood, they seem to breath better than plastic. They are square, normally at least 1200mm and the same height. This gives a good area to fill and spread out the grass cuttings. A bank of 3 I have found to be the best setup, one you are filling, the second is left for the whole of a year and the third is being emptied and used on the garden.