Azure WebJobs SDK now supports instance methods. Combining this with a custom IJobActivator allows you to use DI.
First, create the custom IJobActivator that can resolve a job type using your favourite DI container:
public class MyActivator : IJobActivator
{
    private readonly IUnityContainer _container;
    public MyActivator(IUnityContainer container)
    {
        _container = container;
    }
    public T CreateInstance<T>()
    {
        return _container.Resolve<T>();
    }
}
You need to register this class using a custom JobHostConfiguration:
var config = new JobHostConfiguration
{
    JobActivator = new MyActivator(myContainer)
};
var host = new JobHost(config);
Then, you can use a simple class with instance methods for your jobs (here I'm using Unity's constructor injection feature):
public class MyFunctions
{
    private readonly ISomeDependency _dependency;
    public MyFunctions(ISomeDependency dependency)
    {
        _dependency = dependency;
    }
    public Task DoStuffAsync([QueueTrigger("queue")] string message)
    {
        Console.WriteLine("Injected dependency: {0}", _dependency);
        return Task.FromResult(true);
    }
}
Hope it helped!