How to implement a Least Frequently Used (LFU) cache?

You might benefit from the LFU implementation of ActiveMQ: LFUCache

They have provided some good functionality.

I think, the LFU data structure must combine priority queue (for maintaining fast access to lfu item) and hash map (for providing fast access to any item by its key); I would suggest the following node definition for each object stored in cache:

class Node<T> {
   // access key
   private int key;
   // counter of accesses
   private int numAccesses;
   // current position in pq
   private int currentPos;
   // item itself
   private T item;
   //getters, setters, constructors go here
}

You need key for referring to an item. You need numAccesses as a key for priority queue. You need currentPos to be able to quickly find a pq position of item by key. Now you organize hash map (key(Integer) -> node(Node<T>)) to quickly access items and min heap-based priority queue using number of accesses as priority. Now you can very quickly perform all operations (access, add new item, update number of acceses, remove lfu). You need to write each operation carefully, so that it maintains all the nodes consistent (their number of accesses, their position in pq and there existence in hash map). All operations will work with constant average time complexity which is what you expect from cache.