This is a really useful thing to implement! Good choice of projects. Here are some thoughts on how you could improve it:

#Naming
Generally when I think of a linked list I don't think of a list of links. I think of a list of nodes which are linked to each other (via next and/or previous pointers). For  that reason, I wouldn't call your class `Link`, I'd call it `Node`. The link in your class is the `next` pointer.

The distinction between `push_back()` and `insert()` is unclear. In the base class there doesn't appear to be a difference as `insert()` simply calls `push_back()`. If I make an `LRU` object and call `push_back()` what are the consequences? (Or `push_front()` for that matter?) Perhaps `LRU` shouldn't make the methods of `LinkList` public. Also, renaming `insert()` to `add_new_cache_item()` would make clear which a caller should be using given a particular use case.

Related method names should follow the same pattern. You have `insert()` but then its opposite is called `deleteKey()`. Presumably that's because `delete` is a keyword so you couldn't call it that. I'd either rename `insert()` to `insertKey()` or rename `deleteKey()` to simply `remove()`.

#Do you need `setValue`?
Is there a reason why you need a `setValue()` method in `Link`? It's only called from the constructor. You could simply move the code for it into the constructor and get rid of the public function. (If you added a `next` parameter to the constructor you could make the class immutable which is helpful for multithreading, though it would involve changing some of the logic of updating the list.)

#Encapsulation
I don't generally like `friend` functions. People will say that `operator<<()` is always a `friend` function. But here you're careful to keep the `Link` parameter `const` and it only calls the (also `const`) `getValue()` method. There's nothing  here that requires it be a `friend` function, so I say get rid of the `friend` designation. Just make it a free function in the header.

#Double-ended Queue
What you've created actually has a name other than "linked list". It's often called a [double-ended queue](https://en.wikipedia.org/wiki/Double-ended_queue) or `deque` for short. They have several useful characteristics such as being O(1) for insertion and deletion at the ends, as you've no doubt discovered. There is a standard container called `std::deque` that implements this. (Though I realize you tagged this as "reinventing the wheel", which is always good for learning!)

#Avoid raw pointers
I agree with others that you could implement this with a `vector` or `array` (or `deque`) if you wanted to. But whatever you do, you should avoid using raw pointers. They have so many potential pitfalls that they're really not worth it. I recommend using either `std::unique_ptr` or `std::shared_ptr` depending on the situation. They help avoid a large class of resource management errors.

#Symmetry
You should strive to make the interface for your classes symmetrical to make them easier to use and understand. By that I mean that if you have an `insert()` method that does one thing, the `remove()` or `delete()` method should do the opposite. In `LRU` you've overridden `insert()` to do things like increment the `max_len` counter. But then in `access()` you have to manually decrement it after calling `deleteKey()`. You should also override `deleteKey()` to call the base class and decrement `max_len`  so that someone updating `access()` doesn't have to know this fact in the future. They can simply call `deleteKey()` and not worry about it.