I use this function http://goo.gl/Luxkx to create Fetching Posts by Category with jQuery , but the problem is that I need pagination. For example, I have in some categories more than 10 posts. The first 6 posts show up okay, but I need pagination to show the rest.

You can see how it works now on my site: http://sentisoft.com/bwb/

By the way, for pagination I use code from plugin named WP-PageNavi:

<div class="navigation">
<?php
include('includes/wp-pagenavi.php');
if(function_exists('wp_pagenavi')) { wp_pagenavi(); }
?>
</div>
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Fetching Posts by Category in WordPress with jQuery and pagination

I think's best you post this question on WordPress forum because it's their platform and product.

You are trying to modify a template.

WordPress has a forum where you can post this question.

I'm pretty sure you can find the same question you are having on WordPress and get a straight answer.