How to get to the easternmost Goddess Chest in Skyloft

Those chests are scattered all over the world above the clouds, and can only be opened by activating the corresponding Goddess Cube in the the world below the clouds with your Skyward Strike. Once you do that the chest changes colour and can be opened. When a chest has been activated by a cube, it will show up on your map as a rectangular shape .

This is a Goddess Cube:

Here is a map containing the location of two activated chests (indicated by the rectangular blue symbol):

Here is a picture of an activated chest, which is now able to be opened:

(images taken from: http://www.ign.com/wikis/the-legend-of-zelda-skyward-sword/Goddess_Cubes)

You can get Monster Claws from Keese (bats) of any type. Just look for a place that has a lot of them and grind it out.

A place that comes to mind at that point in the game where you can find a lot would be the Waterfall Cave. That's the cave in Skyloft which is behind the waterfall where you had to go get your Loftwing. It should be relatively easy to go through that multiple times.

If you can wait a little bit longer, you might want to get the Treasure Medal before doing this to give them a better chance at appearing. You'll get this when you reach Eldin Volcano from one of the Goddess Cubes.

To the east of the Fire Temple behind the houses down by the lava.