Class contains private constructor, and public property, then how can we assign the public property in C#

Private constructors are used when the class needs to maintain control over the instances that are created (most often as a Singleton class where there is only ever one instance of the class created).
In those circumstances, the class needs to provide a method to create and return an instance, or the non-static properties and methods cannot be accessed at all (since they require an instance in order to be used).

So in your specific example, the answer is "you can't".
But you can if you provide a method as mentioned:

public class mUIPassword
    {
    public mUserPasswords user { get; set; }
    public string channel { get; set; }
    public string lang { get; set; }
    private mUIPassword()
        {
        channel = "device|Web";
        lang = "en";
        }
    private static mUIPassword theInstance = null;
    public static mUIPassword GetInstance()
        {
        if (theInstance == null) theInstance = new mUIPassword();
        return theInstance;
        }
    }