Grass is always going to struggle to grow and establish a good root system under a large established tree, especially if there is lots of leaf cover. If you wish to keep the tree, have you considered pruning it some to allow more light get to the ground?
You need to prepare well for sowing seed, this involves breaking up any clogs of soil with a fork and allowing air to get deeper down into the soil. What type of soil have you got? Loaf, Clay? Get to know your soil and how compacted it gets. If it is compacted and gets wet and stays wet in winter, this means it may have a large clay content, which incorporating organic matter will help to improve drainage.
Once you are happy with the ground and ready to sow, firm the ground by walking on it to even it out of any lumps and low laying areas. A rake help to level it out. You can always put some topsoil (medium soil with lots of fine grit or sand) on to provide a good base for the grass seed. Rake this to a fine bed. You can now consider applying a pre-sowing fertiliser to promote root growth. Then sow the seed evenly.
For a shady lawn consider a mixture with perhaps Lesser Timothy with Velvet Bent grass, or maybe a dwarf Perennial Ryegrass.
It will take about 2-3 weeks for germination and try not to walk on it, keep it watered but not too wet or it will rot. The first mow should only trim the very top. Perhaps about 1 & 1/4 inch tall for the first cut. Also look out for early signs of diseases (yellowing is a sign, brown patches etc...) and keep moss at bay.
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