This depends on various factors.
Extbase Extensions
If the Extension is based on Extbase, you could write your own Extension with your custom Controller and use the domain model of the original extension. Since you didn't really specify which extension you want to modify, this is only a general approach.
Hooks
Are not necessarily provided by extensions. You can always ask the extension author to provide a new hook.
XCLASS
With XCLASS, you can overwrite a class from your own extension. You can find more information about this here. If you update the original extension, you probably need to adjust your XCLASS code.
Directly modify an existing Extension
You should avoid doing this. But if it is your only option, you can modify the file "ext_emconf.php" of the extension and set "state" to "excludeFromUpdates":
$EM_CONF[$_EXTKEY] = array(
'title' => 'Extension Title',
'description' => '',
'category' => 'plugin',
'state' => 'excludeFromUpdates',
...
);
This excludes an extension from updates.