Actually, all the guild for how to use Umbraco Api is here. But the key is to create a controller in folder App_Code of deployed version of Umbraco CMS; the controller then must be inherited from UmbracoApiController.

There is an example:

public class ProductsController : UmbracoApiController
{
    Product[] products = new Product[]
    {
            new Product { Id = 1, Name = "Tomato Soup", Category = "Groceries", Price = 1 },
            new Product { Id = 2, Name = "Yo-yo", Category = "Toys", Price = 3.75M },
            new Product { Id = 3, Name = "Hammer", Category = "Hardware", Price = 16.99M }
    };

    public IEnumerable<Product> GetAllProducts()
    {
        //Connect to the Umbraco DB
        var db = ApplicationContext.DatabaseContext.Database;

        //Get an IENumberable of Product objects to iterate over
        var products = db.Fetch<Product>(new Sql().Select("*").From("Products"));

        return products;
    }

    public IHttpActionResult GetProduct(int id)
    {
        //Connect to the Umbraco DB
        var db = ApplicationContext.DatabaseContext.Database;

        //Get an IENumberable of Products objects to iterate over
        var product = db.Fetch<Product>("SELECT * FROM Products WHERE Id=@0", id).FirstOrDefault();

        if (product == null)
        {
            return NotFound();
        }
        return Ok(product);
    }
}

All api url from Umbraco should start with “api” and then Controller Name “Products” and then Method Name “GetAllProducts” Example: “api/Products/GetAllProducts” Example Ajax:

var uri = 'api/Products/GetAllProducts';

$(document).ready(function () {
      // Send an AJAX request
      $.getJSON(uri)
          .done(function (data) {
            // On success, 'data' contains a list of products.
            $.each(data, function (key, item) {
              // Add a list item for the product.
              $('<li>', { text: formatItem(item) }).appendTo($('#products'));
            });
          });
    });

Note: For how to create a helloworld plugin for Umbraco can be seen here.