Reusable Ajax object
From a quick once over:


JsHint.com has very little feedback, good!
Consider using use strict to activate strict mode
Avoid console.log
Both beforeSend , error and complete should be customizable by the caller
success() returns ( PlainObject data, String textStatus, jqXHR jqXHR ) you should pass all 3 to success_callback
JavaScript is lowerCamelCase, success_callback -> successCallback
member_id: 1, &lt;- Maybe this is sample code, this magic constant deserves a comment
I wonder if AjaxRequest should know about "/ajax-member/, from a DRY perspective
for (i in response.achievements) {, response.achievements seems to be an array, please use a regular for loop to iterate over arrays, it will prevent hard to find bugs at a later stage
Find a good templating library, you need it, consider Handlebars
I think your Ajax class could be expanded, getAchievements and getComments are clearly a result of copy pasting code