There are several aspects of this:
Give feedback to the user that the item has been added to the cart
Let the user continue browsing your store and buy more items
Prevent the user from accidentally buying an item twice
If you look at Amazon, they generally navigate to a separate page that tells you the item was added, and also shows suggestions of what to buy next. That covers #1 and #2, and since the "buy" button is not on the new page, also #3. They also have a popup where you can choose to buy several copies of something, right above the "buy" button. Also, there's a button that lets the customer buy now whatever is in the cart so far.
However, I often find the change of page annoying when there were suggestions on the previous page I also wanted to buy. An alternative would be:
For a short while (at least until the request is through) disable the "buy" button. Then show a temporary banner at the top of the page that says "Item added to cart" and after a short delay, reactivate the button.
If the user clicks the button again, check if the item is already in the cart, and if it is, ask for confirmation and how many times it's been added so far, and that this will add it another time.
Also: Many web sites have a shopping cart icon somewhere prominently, which displays the number of items in the cart on it, and quickly lets the user get to the "buy now" page. It is very important that the user gets to see where they can go after they've added an item to the cart to actually buy it. Maybe you'd want to add a "go to checkout" button somewhere to the confirmation banner mentioned above to make this obvious.