How do I set a readonly field in an initialize method that gets called from the constructor?

Rob's answer is the way to do it, in my book. If you need to initialize multiple fields you can do it using out parameters:

public class Class
{
    private readonly int readonlyField1;
    private readonly int readonlyField2;

    public Class()
    {
        Init(out readonlyField1, out readonlyField2);
    }

    protected virtual void Init(out int field1, out int field2)
    {
        field1 = 1;
        field2 = 2;
    }
}

Personally I find this makes sense in certain scenarios, such as when you want your fields to be readonly but you also want to be able to set them differently in a derived class (without having to chain a ton of parameters through some protected constructor). But maybe that's just me.

Instead of using an Initialize method, how about inheriting a basic constructor through all your other constructors. i.e.

public class MyClass
{
    readonly int field1;
    readonly double field2;
    public MyClass(int field1, double field2)
    {
        //put whatever initialization logic you need here...
        field1 = 10;
        field2 = 30.2;
    }
    public MyClass(int field1, double field2p1, double field2p2)
        : this(field1, (field2p1 + field2p2))
    {
        //put anything extra in here
    }
}

This may be a little late to reach the original person in need, but it seems like this will cleanly solve the problem... Without the need to use any sort of nasty reflection or out parameters.

The only solution I can think of is to return the value from the Init() method that the readonly field needs to be assigned:

public class Class
{
    private readonly int readonlyField;

    public Class()
    {
        readonlyField = Init();
    }

    private int Init()
    {
        return 1;
    }
}