To unlock Princess Peach you have to complete the world tour which includes 6 worlds, each consisting of four levels. Note that you must purchase the game first to be able to access all levels.
You can unlock Yoshi when you build Yoshi's House. You will get access to the house in kingdom builder once you collect 30 Red Toads and 30 Yellow Toads in the Rally mode. Before getting into a Rally, make sure the audience includes Red or Yellow Toads.
Toad is the simplest character to unlock. To unlock Toad you just have to Link your Nintendo account to the game. See why Super Mario run is a good game.
To unlock Luigi you will have to collect 150 Green Toads, and 150 Purple Toads in the Toad Rally mode. Before getting into a Rally, make sure the audience includes green or purple Toads.
You can unlock Toadette by purchasing Toadette's House in the kingdom builder. You will need to collect 200 Red Toads, 200 Blue Toads, and 200 Green Toads in the Toad Rally mode. Toadette is the most difficult character to unlock.