It's simple. When you register the constructor, you just pass the value you want injected for the parameter. The container matches up your constructor based on the type of value (API) or name of parameter (XML).
In the API, you'd do:
container.RegisterType<MyType>(new InjectionConstructor("My string here"));
That will select a constructor that takes a single string, and at resolve time will pass the string "My string here".
The equivalent XML (using the 2.0 config schema) would be:
<register type="MyType">
<constructor>
<param name="whateverParameterNameIs" value="My string here" />
</constructor>
</register>