Create filter posts by custom field checkboxes

I am using a custom post type with custom fields. One custom field contains 10 checkboxes with 10 values. I would like to display these checkboxes on the custom post query and allow the user to filter down the posts by checking the values.

What is the best method to accomplish this? The only way I can think of to filter a post by a single value is to switch to using categories. is it possible to filter posts with a dropdown menu or an array of checkboxes with the values defined in the custom field?