i have a real problem with moss in my lawn due to poor drainage because of our clay soil.
i treat the lawn with a weed feed and mosskiller product in early april, i wait a month then rake all the dead moss out, i aerate the lawn by spiking it and then re-seed and top dress the whole lawn.
i repeat this process in early autumn, applying the weed feed and mosskiller the first week of september.
be warned that the first time you do this, depending on how much moss there is compared to grass, your lawn will look awful when you've raked all the moss out, i thought to myself christ what have i done, bald patches everywhere,it will look great once the new seeds took hold though.
don't do anything about the moss at this time of the year because you will struggle to grow new grass because of the dry weather, thats why the spring and autumn is best, it offers perfect growing conditions.
and lastly, in the autumn you need an autumn formula weed feed and mosskiller, which is different to the spring one.