Some methods you can try:
A) First thing to try is pouring pans of boiling hot water which should dissolve any grease and solids and move them along and out of the pipework.
B) Using a sink plunger. You need to cover the overflow with a cloth and then place the plunger over the sink hole and “work” the plunger vigorously. The plunger action creates pressure and sometimes is enough to unblock it.
C) A Chemical unblocker such as Sulphuric Acid or Mr Muscle (you can buy a powerful one from a plumbers merchants for around £10). PLEASE REMEMBER TO WEAR SAFETY EQUIPMENT.
D) You can remove the trap under the sink and clean out with wire. Take the opportunity to clean the sink hole from underneath.
Good idea to remove the flexi hose that attaches to the sink overflow and flush out by attaching to a tap. You will be amazed at what gunk flies out!
E) With the trap removed (and cleaned) you can use a drain snake (or curtain wire) into the open waste pipe to see if you can dislodge any debris.
F) If possible dismantle ALL the waste pipework and renew or clean out.
G) If you have tried all of the above and are still having problems, you will need to look at your underground drainage. You should look for a manhole and try rodding the drains.