var j$ = jQuery;
var search_text = j$(".cv__text").val();
j$.post({
    url: "<?php echo admin_url('admin-ajax.php'); ?>",
    data: ({
        action: "search_cv", search_text: search_text
    }),
    success: function (response){
        console.log(response);
        j$("#search_results").html(response);
    }
});

function search_cv(){
    $search_text = ucfirst($_POST["search_text"]);

    //Creating New DB Connection

    $database_name = "employee_cv";

    $mydb = new wpdb(DB_USER, DB_PASSWORD, $database_name, DB_HOST);
    $mydb -> show_errors();

    if ($_POST["search_text"] != null){

        $result = $mydb -> get_results(
            $mydb -> prepare(
                'SELECT * FROM employee
                WHERE Experience = %s',
                $search_text
            ) );

    }

    foreach ($result as $employee){
        //Here I just echo html blocks (divs, paragraphs etc.)
    }

    die();
}

add_action( 'wp_ajax_search_cv', 'search_cv' );
add_action( 'wp_ajax_nopriv_search_cv', 'search_cv' );

<form class="searchCV_form" role="form" action="">
    <div>
        <input type="text" id="search_text" name="search_text" class="cv__text">
        <span class="input-group-btn">
            <button type="submit" class="btn btn-default btn-primary cv__button search--form-btn">SUBMIT</button>
        </span>
    </div>
</form>
<div id="search_results"></div>

$("form.searchCV_form").submit(function(event){
    event.preventDefault();

    alert( "We're not going anywhere now..." );
    // j$.post...
});

$("form.searchCV_form").submit(function(e){
    alert( "We're not going anywhere now..." );
    // j$.post...

    return false;
});

<form class="searchCV_form" role="form" action="">
     <div>
         <input type="text" id="search_text" name="search_text" class="cv__text">
         <span class="input-group-btn">
             <button type="submit" class="btn btn-default btn-primary cv__button search--form-btn">SUBMIT</button>
         </span>
     </div>
 </form>
 <div id="search_results"></div>

<script>
// wrap everything in a closure
(function($){

  // get our references
  var $form = $('form.searchCV_form'),
      $search_field = $('#search_text'),
      $results = $('#search_results');

  // AJAX search call
  function do_search() {

    // grab the query value from the search field
    var search_text = $search_field.val();

    // do a POST ajax call
    $.ajax({
      type: "POST",
      url: '<?php echo admin_url('admin-ajax.php'); ?>',
      data: ({
        action: "search_cv",
        search_text: search_text
      }),
      success: function (response){
        console.log(response);
        $results.html(response);
      }
    });
  }

  // on submit, do the search but return false to stop page refresh
  $form.submit(function(e) {
    do_search();
    return false;
  });

})(jQuery);
</script>

function search_cv()
{
    // get the search query
    $search_text = ucfirst($_POST["search_text"]);

    // clean it up
    $search_text = sanitize_text_field( $search_text);

    // ... do stuff with it

    // output the HTML which will be consumed by $.html()
    ?><div>You searched for <?php echo $search_text; ?> and we found... </div><?php

    // stop doing stuff
    die();
}

add_action( 'wp_ajax_search_cv', 'search_cv' );
add_action( 'wp_ajax_nopriv_search_cv', 'search_cv' );

url: "<?php echo admin_url('admin-ajax.php'); ?>",

// Register the script
wp_register_script( 'some_handle', 'path/to/myscript.js' );

// Localize the script with new data
$translation_array = array(
    'myajax_url' => admin_url('admin-ajax.php'),
);
wp_localize_script( 'some_handle', 'object_name', $translation_array );

// Enqueued script with localized data.
wp_enqueue_script( 'some_handle' );

url: some_handle.myajax_url,

// Register the script - located in functions.php (usually)
wp_register_script( 'some_handle', 'path/to/myscript.js' );

//In the file you need the ajax results

// Enqueued script
wp_enqueue_script( 'some_handle' );

// Localize the script with new data
wp_localize_script( 'some_handle', 'object_name', array('ajax_url' => admin_url( 'admin-ajax.php' ) );

// the ajax call in the js file
jQuery.ajax({
    url : object_name.ajax_url,
    type : 'post',
    data : {
           action:'search_cv',
           search_text: search_text
    }
    success : function( response ) {
         response = jQuery.parseJSON(response);
         //use the response variable to print what you need on the webpage. Basically instead of running the foreach in the php function run it here with js
   }
 });

echo json_encode($results);