When you read a Bond price in the newspaper, on a web site, in a database of bond prices it is always the Clean Price. [You don't have to compute anything! The clean price is there!]. When you actually buy the bond you receive an invoice asking you to pay the Clean Price plus the Accrued Interest, which are added together for your convenience and are called the Dirty Price.
It is similar to a restaurant, where a hamburger is listed for 1,99 EUR but when you get the bill at the end of the meal there is is a service charge,a tax, and maybe other unexpected items which bring the bill to 2,07 EUR.
The service charge compensates the waiter who brought the meal to you, the accrued interest compensates the seller of the bond who is ethically entitled to a portion of the next coupon you will receive (if he held the bond for a part of the coupon period, for example if he held for 1/2 the coupon period he is entitled to half the next coupon under accounting "accrual" principles). Essentially the accrued interest is a mechanism for sharing the value of the next coupon (which the buyer will receive) in a fair way between buyer and seller based on when in the coupon period the bond changed hands.