How do you programatically add a user to a user entry field
You will need a custom plugin to do this.

I'll assume that you're familiar with that process. 

You can build a form in your template like this, passing in the entry ID you want to add the user to and the user ID you want to add.

&lt;form method="post" accept-charset="UTF-8"&gt; 
    {{ getCsrfInput() }}
    &lt;input type="hidden" name="action" value="myPlugin/save"&gt;
    &lt;input type="hidden" name="redirect" value="/success"&gt;

    &lt;input type="hidden" name="elementId" value="{{ entry.id }}"&gt;

    &lt;input type="hidden" name="userId" value="{{ currentUser.id }}"&gt;

    &lt;button type="submit" &gt;Submit&lt;/button&gt;
&lt;/form&gt;


Then, you'll need a controller action in your plugin to which you submit a POST request from a form.

public function actionSave()
{
    $this-&gt;requirePostRequest();

    // get all the post variables
    $elementId = craft()-&gt;request-&gt;getPost('elementId');
    $userId = craft()-&gt;request-&gt;getPost('userId');

    // set the name of the field you want to update
    $fieldHandle = 'myUserField';

    // based on the element and field vars, get the models for each
    // using the elements and fields services
    $element = craft()-&gt;elements-&gt;getElementById($elementId);
    $field = craft()-&gt;fields-&gt;getFieldByHandle($fieldHandle);

    // store the current ids into a variable
    $currentIds = $element-&gt;{$field-&gt;handle}-&gt;ids();

    // merge the new id into the current ids
    $newIds = array_merge($currentIds, (array) $userId); 

    // save the new array of ids, using the relations service
    craft()-&gt;relations-&gt;saveRelations($field, $element, $newIds);

    // redirect to the redirect url
    $this-&gt;redirectToPostedUrl();

}


EDIT:

It's very important that you follow the naming conventions in the docs. 

You controller class should be named MyPluginController, which would then make the action in the form myPlugin/save (controller and action get removed from the exposed plugin and action names). 

Also the redirect should be a page on the side where you want the user to go after being added, not pointing to your plugin.