Wrap your search form and grid in pjax. When clicking search it will update both. There are some wikis in the wiki section on this.
If you want to submit the form on click of a checkbox you will need to use the pjax along with JS/jQuery to detect the change and submit the form. There are a lot of ways to do this… just search something like ‘submit form on check of checkbox jquery’ in google.
If you still need help after looking up and trying some ways… post your form and gridview code and what you tried.