Using a trigger to update the view could be a big performance hit. You should never use triggers unless absolutely necessary. They are monsters that drag your performance down.
When you use
WITH SCHEMABINDING you must specify the schema along with the tablename otherwise you get the following error:
Cannot schema bind view 'viewXXX' because name 'tableXXX' is invalid for schema binding. Names must be in two-part format and an object cannot reference itself."
If you use two-part-naming and your script has to be run on multiple databases you will run into a problem when there are different Schemas across databases. In that scenario you have to use dynamic SQL to create the scripts.
So, if you're dealing with just one database then use the tablename with the schema like
Schema.TableName. When I says two-part-naming this is what I mean.