Sink or Swim 5
Dive into the Money Bin
After finishing the tutorial level you will be in Scrooge's office. To the right there will be the level select screen, to the left is the Art Gallery and in the center will be the vault. Choose to go into the vault and simply jump into the gold. The achievement will pop after a few seconds.
Score to Settle 10
Scrooge doesn't take kindly to being swindled by statues
In the Amazon level towards the end there will be a lone statue on the right of a rope to climb. This statue in the original NES version asked you for money to pass so in the remake you will get a bit of revenge. Pogo jump to the top of the statue and destroy it. You will be rewarded with a nice chunk of change and an achievement. If you refer to the Bless Me Bagpipes video guide, the statue is shown around the 1:09 mark in the video on the right before Scrooge goes up the rope.
A Few Gears Loose 10
Get to the secret gyro area in the clouds
This seems to be random as to when you get there. I have gotten to the cloud area after choosing my first stage in some playthroughs but I've also had an entire playthrough without getting it once. Just keep playing and you will find this area. You are taken to the clouds at the start of a level, just continue through collecting the gems or fall. Either way the achievement will pop for you.
It's A Duck Blur 10
Defeat 5 enemies under the effect of 1 magic coin
You can easily get this on most levels but the Amazon and Moon levels seem to be the easiest. Once you get the magic coin simply run into 5 enemies before it wears off. Note that this may only be obtained on Easy or Normal difficulties since the higher difficulties do not have the magic coins.
Healthy Appetite 5
Eat some of Mrs. Beakley's delicious treats.
You will find Mrs. Beakley in pretty much every level. Just make sure to explore every area that you can and you will easily find her. It's good to get an idea of where she is so that on harder difficulties you can earn extra health goodies to help you along.
Pogo A Go Go 15
Defeat 5 enemies in a row with pogo jump without touching the ground
Simply pogo jump on 5 enemies in a row. If you are having trouble with this don't worry too much since there is a section in the Vesuvius level that requires you to jump on 5 enemies to pass a certain part and the achievement will unlock for you then.