Hi , I have purchased an old fish and chip shop, do I need self closing fire doors leading from the frying area at the front of the shop leading into the kitchen at the back ?

Not necessarily (Your FRA should determine this) Many smaller fish & chip shops have such short travel distances there is no need for fire separation for life safety/compliance purposes and often have open doorways between the two areas.

The only complication with these premises is if there is upstairs living accommodation accessed from inside the shop.