For now mow it regularly, and on quite a low setting (work down to it if you dont already mow low).
Come autumn use some Evergreen 4 in 1 or Iron Sulphate based feed/weedkiller.. Follow instructions exactly and err on the side of too little (two much and it kills EVERYTHING). Apply it, wait... Weeds will die, moss will die and grass will grow!! Borrow a scarifier if you can, if not use a thin tithe rake and prepare for some work.. Rake it longways, lifting all the dead stuff out and binning it (DONT compost heap it, moss spores etc). Then do the same again 45 degrees different.. and again, another 45 degrees different.
Now get a fork and aerate the soil, lots of holes. Water it lots. Liberally apply grass seed, then top dress with compost (over the grass seed), water it. Not a light watering but saturate it.. gently though, dont want to wash the seed away. Now walk all over it to compress the compost/seed mix together.
Water daily if the weather doesnt do it for you.
It wont take long to sprout, let it grow a bit (2-3 leaves per seed) then mow it on a high setting. over the next few weeks gradually lower.
You will have missed bits, some wont germinate and some will be eaten by birds.. Especially if its a big area/lots of weeds (when we did our lawn it was over 70% weeds/moss... :EEK: !!), rinse and repeat the seed and compost steps.
Once its growing well stop watering it so often, let it have a week without water (if weather allows), keep an eye, if its hot and starting to dry out/look stressed then do water it BUT you want it to grow good roots to get its water from further down.
Its not rocket science, but can be a pain when it goes wrong.. BIGGEST mistake I made was overusing the weed and feed stuff, killed big patches of grass.
Good luck!