In almost all cases, just putting down new grass seed (after painting the lawn or not) will rarely solve your problem because the new grass, unless lawn was killed by under watering or lack of fertilizer/lime or because it was the wrong type of seed for the area or had one-time chemical damage, will do no better than the old unless the growing conditions are fixed first. First step is a soil analysis - many greenhouses and home centers can do this, as will your local Cooperative Extension Service. CES can also help you with advice on which grasses grow best in your area, and if you take a divot in can help with identifying what species you have now.
Sodding is almost always most expensive, but most immediately effective because you are importing living grass with its own root biosphere - you KNOW the grass mix and soil are correct, and if placed over a new topsoil layer can have higher survivability in dry areas and higher traffic areas because the grass is not trying to get started under difficult conditions. Cost varies greatly depending on area and how easily turf is grown there - can be from $0.25-$2.00/SF, but generally around $1.00/SF plus or minus $0.25/SF in many areas.
Hydroseeding is spraying a mix of mulch (usually shredded lumbermill waste cellulose fibers or shredded newspaper), fertilizer, dye, and grass seed - is colored green or bluish to make it look a bit like grass up front, to show it has been hydroseeded so people hopefully keep off it, and so operator can easily see where he has already sprayed and how thickly.
Hydroseeding over a 1-2" topsoil layer (thicker in some extremely dry areas where more topsoil is needed to hold the moisture, and in some areas with nutritionally deficient soils like dunes, sandy/gravelly river deposits, clays) is generally the cheapest commercial method. Good for new lawn on flat or gentle slopes, requires additional erosion protection like a mulch blanket on steeper slopes. Not very effective on existing lawn unless the chemistry is fixed or a thick topsoil layer is put down, for same reason as with overseeding. Costs depends on how common it is in your area, but generally in the $0.10-0.15/SF range for a typical residential whole-yard job - down to as little as about $0.07/SF for areas over about an acre, in the $0.15-0.20/SF for small areas like 1000 SF or less. This is for the mulch and seed and spraying only - no topsoiling or surface prep, but you need that done in each case regardless.
Cheapest method is figuring out why existing lawn failed, fix grass mix or soil chemistry, either thatch and/or core or add a thin layer of topsoil (1/2-1" typically) - whichever is needed then seed and, on steeper slopes cover with biodegradable mulch blanket. Typically about $0.05-0.10/SF for do it yourself seeding materials, not including topsoil or fertilizer, which is needed for each case regardless.
Topsoiling can run from about $10-40/CY depending on how available in your area and quantity purchased (truckloads to pickup load range above) for the dirt - $15-20 is a common number by the full 12-20CY truckload. Spreading and raking out typically about $50-100/CY additional for small lawns, $30-50/CY for larger areas like estate lawns and golf courses.
Painting grass is rarely done - actual paint is bad for the grass and especially for new seed but there are vegetable based dyes that are relatively harmless and last a few months. Basically a cosmetic procedure done right before a big party or barbecue or right before house listing or such to spiff your lawn up. Also being done in some water shortage areas with severe restrictions or bans on lawn watering. Cost from about $0.10/SF doing it yourself, to $0.15-0.25/SF depending on lawn size to have done commercially.
Landscaping company is what you want for complete job, or if you are going to do most work yourself, then sod supplier or hydroseeder those two items only. Hope this answered your question - if not, reply back using the Answer This Question button under your question.