Basic example of a WebApi Controller getting a ContentService in the constructor.

WebApiUmbracoController.cs

using System;
using System;
using System.Net.Http;
using System.Web.Http;
using Umbraco.Core;
using Umbraco.Core.Services;

namespace Example.Controllers
{
    public class WebApiUmbracoController : ApiController
    {
        private IContentService _contentService;
        
        public WebApiUmbracoController()
        {
            _contentService = ApplicationContext.Current.Services.ContentService;
        }
    }
}