Unless you have a grass that can crowd out weeds, I would use the herbicide or dig out most of the persistant weeds.
If the ground is very compacted, I would kill everything and roto-till 6-8 inches, add compost fertilizer and an irrigation sytem if you don't already have one, unless watering is not an issue for you. An irrigation system is a plus when you sell.
Select a grass that competes well with the weeds and give it the best start possible. You can do this in stages, but you will need to keep up with weeds in the entire lawn or they will just come back.
Make sure the new lawn is maintained at the proper height and that you give it enough water and aerate, dethatch, topdress and fertilize.
In mild climates without snow, fall is the best time to start a lawn. The grass won't grow much up top but the roots will get a good start and they will be able to take the summer drought better.
I have not started a lawn from seed, only stolons, plugs and sod. Most grass seed sown here either is not appropriate (shady lawn mixture containing Fescue and Kentucky blue grass that don't do well in the tropics), centipede grass, or burmuda grass. The most common grasses here are zoysias which compete with weeds well, are fairly drought tolerant, and army worms, fungal diseases, and maintenance are not as much of a problem as with the bermuda hybrids. Zoysia grows slower and always from plugs or stolons, but you can get away with mowing every two weeks instead of every week. All lawns require a lot of water to look green. Most lawns will go dormant and yellow in summer but come back once the rains come.
You should select the type of grass that best fits your location, competes well with weeds, and is easy to maintain.
Happy gardening in Hawaii. Gardens are where people grow.