Files security - files folder hacked
From the information given, it is impossible to give a full answer.  For instance: Are your site on a dedicated server, or on shared hosting?  What user and group "owns" the files in your Drupal installation?  What is the extension (like .php) of the unwanted files somebody places in the imagecache subfolders?  (Please edit your question and expand it with this information, it will be a great help in order to give a more precise answer.)

However, since most people use shared hosting, I'll assume that you're using that.  You can read about how to set up correct ownership and permissions for in a shared hosting environment in this article: What are the most appropriate users and permission levels for Drupal sites on shared hosting?

However, the permissions you cite, and in particular the fact that 750 "breaks the website" indicate that you must have gotten something (probably the ownership to the files in your Drupal installation) wrong.  750means rwxr-x--- while 755 means rwxr-xr-x. 

In a shared environment, the web server will either:


Run as the owner of the site (i.e. you). In this case your files/ folder (upload directory) should have 700 (rwx-----).
Run as a member of the www group. In this case your files/ folder (upload directory) should have 770 (rwxrwx--).


Not under any circumstance do you want the last triplet (permissions for others) to be anything else than zero.

However, unfortunately, wrong file permissions is probably not the only reason your files/ directory has been compromised.  Both 755 and 750 only gives write permission to the owner of the files/ directory.  You haven't told us who that owner is, but its either you or the web server user.

If it is you, then your shell account has been compromised, so look for Trojans with the excecute bit set (and change your shell account password).  If it is the web server user, then you have a cross-scripting attack or a PHP-injection attack, and you need to find and fix that.