Currently Amazon does not ship the Echo to Singapore.
The only way to get it is through freight forwarders.
Option 1 : Amazon direct
You pay all shipping and other charges. Approx US 180 + 50 SGD shipping
Sign up for Borderlinx
There are other websites that do this but I find them to be the most reliable and quick.
Create an Amazon Account
Not sure what you are doing if you dont have an account :)
Set your address to your Borderlinx account
Everyone's address is unique so be sure to use yours!
Buy the Echo and checkout as normal
Remember to set your borderlinx address as your shipping address
Option 2 : Carousell
You pay a premium and there needs some element of trust. Approx 300 SGD