Here is a youtube video showing how to get the Secret Cake. It is in the Birdday Party section, level 18-15.
Basically what you want to do is fire the first two birds so that some of the blocks fall off the right side and hit the donut lever or the tnt.
Once you have managed to do that, if you look below where you fire off the birds, a chest has now appeared. If you see it, fire the 3rd bird as the video shows and hit the chest. Hopefully you still have some blocks that fall in and hit the tnt crate netting you the achievement. If not, fire your 4th bird and hope you get some in.
It may take a few tries but it does work.
***9/12/15 Update from HolyHa1fDead***
"Sometime since 2013 there must have been an update.
6. Mine and Dine has 3 pages of levels - 15, 16 & 17
7. Birdday Party has 3 pages - 4 Candles, 3 Candles & 2 Candles
8. Bad Piggies also has 3 pages - 21, 22 & 23
It looks like the pages for Birdday Party have been reorded from 18-19-20 to 20-19-18. In otherwords it is now level 15 on the third page (2 Candles) of Birdday Party, instead of level 15 on the first page."
Good Luck and if I can help make the guide better let me know!