How to set a binding in Code?

You need to change source to viewmodel object:

myBinding.Source = viewModelObject;

Replace:

myBinding.Source = ViewModel.SomeString;

with:

myBinding.Source = ViewModel;

Example:

Binding myBinding = new Binding();
myBinding.Source = ViewModel;
myBinding.Path = new PropertyPath("SomeString");
myBinding.Mode = BindingMode.TwoWay;
myBinding.UpdateSourceTrigger = UpdateSourceTrigger.PropertyChanged;
BindingOperations.SetBinding(txtText, TextBox.TextProperty, myBinding);

Your source should be just ViewModel, the .SomeString part is evaluated from the Path (the Path can be set by the constructor or by the Path property).

In addition to the answer of Dyppl, I think it would be nice to place this inside the OnDataContextChanged event:

private void OnDataContextChanged(object sender, DependencyPropertyChangedEventArgs e)
{
    // Unforunately we cannot bind from the viewmodel to the code behind so easily, the dependency property is not available in XAML. (for some reason).
    // To work around this, we create the binding once we get the viewmodel through the datacontext.
    var newViewModel = e.NewValue as MyViewModel;

    var executablePathBinding = new Binding
    {
        Source = newViewModel,
        Path = new PropertyPath(nameof(newViewModel.ExecutablePath))
    };

    BindingOperations.SetBinding(LayoutRoot, ExecutablePathProperty, executablePathBinding);
}

We have also had cases were we just saved the DataContext to a local property and used that to access viewmodel properties. The choice is of course yours, I like this approach because it is more consistent with the rest. You can also add some validation, like null checks. If you actually change your DataContext around, I think it would be nice to also call:

BindingOperations.ClearBinding(myText, TextBlock.TextProperty);

to clear the binding of the old viewmodel (e.oldValue in the event handler).