To implement the change password functionality into your custom portlet in Liferay 6.2, just follow the below steps:

1- Write the below line in liferay-portlet.xml file for the portlet in with you want to perform action to change the password

<private-session-attributes>false</private-session-attributes>

2- Write the below lines in protal-ext.properties to enable the session to store the current password after login

session.shared.attributes.excludes = USER_PASSWORD
session.store.password=true

3- Write the below code to implement the UI for "Change Password" in the jsp file

change_password.jsp

<liferay-portlet:actionURL var="updateUserPassword" name="updateUserPassword">
</liferay-portlet:actionURL>
   <aui:form  action="<%=updateUserPassword%>" id="updateForm" name="updateForm" method="post" autocomplete="off">

   <liferay-ui:error exception="<%= UserPasswordException.class %>"
        message="that-password-has-already-been-used-please-enter-in-a-different-password" />
   <liferay-ui:error key="error-key" message="Current Password is invalid."></liferay-ui:error>
   <aui:fieldset cssClass="lfr-portrait-editor">
         <aui:input autoFocus="<%= true %>" type="password" label="Current Password" value="" name="currentPassword" autocomplete="off">
         <aui:validator name="required" />
         </aui:input>
         <aui:input label="New Password" name="password1" size="30" type="password" value="">
            <aui:validator name="required" />
         </aui:input>

                <aui:input label="Enter Again" name="password2" size="30" type="password" value="">
                    <aui:validator name="required" />
                    <aui:validator name="equalTo">
                        '#<portlet:namespace />password1'
                    </aui:validator>
                </aui:input>
         <aui:button-row>
            <aui:button type="submit"/>
        </aui:button-row>
    </aui:fieldset>
   </aui:form>

4- Finally write the below method in portlet action file

public void updateUserPassword(ActionRequest actionRequest,ActionResponse actionResponse) throws PortalException, SystemException, EncryptorException, IOException, PortletException
    {
        try {

            ThemeDisplay themeDisplay = (ThemeDisplay)actionRequest.getAttribute(WebKeys.THEME_DISPLAY);
            HttpServletRequest request = PortalUtil.getHttpServletRequest(actionRequest);
            HttpSession session = request.getSession();
            //Get the user's current password from session
            String loginPassword = (String)session.getAttribute(WebKeys.USER_PASSWORD);

            String currentPassword = (String) actionRequest.getParameter("currentPassword").trim();
            String newPassword = (String) actionRequest.getParameter("password1").trim();
            String confirmPassword = (String) actionRequest.getParameter("password2").trim();

            if (loginPassword.equals(currentPassword) && newPassword.equals(confirmPassword)) {
                UserLocalServiceUtil.updatePassword(themeDisplay.getUserId(), newPassword,confirmPassword, false);

                //Set the entered new password into session after changing the password
                session.setAttribute(WebKeys.USER_PASSWORD, newPassword);

                SessionMessages.add(actionRequest, "request_processed", "Password updated successfully.");
                actionRequest.setAttribute("passwordUpdated", "updated");
                actionResponse.setRenderParameter("mvcPath", "/change_password.jsp");
            } else {
                SessionErrors.add(actionRequest, "error-key", "Sorry, Something wrong.");
                actionResponse.setRenderParameter("mvcPath", "/change_password.jsp");
            }

        } catch (UserPasswordException e) {
            SessionErrors.add(actionRequest, UserPasswordException.class.getName(), e);
            actionResponse.setRenderParameter("mvcPath", "/change_password.jsp");
        }
    }

Hope this will help you :)