How to get CPT category checkbox list and show post of selected(multiply) checkboxes via ajax?

Try adding return false; to prevent the default submit action of the click.

$('.grp_getnew').on('click', function(){
    var data = {
        action: 'get_random_post_tu',
        usecategory: $('#categoryselect').val()
    };

    $.post( ajax_object.ajax_url, data, function(response) {
        if ( response.status != 'error' ) {
            var $link = $("<a href="" + response.link + "">" + response.title + "</a><span >" + response.content +"</span>");
            $('.grp_content').html($link);
        }
    }, "json");
    return false;
});