Cannot remotely connect to SQL Server 2012 VM on Azure
basic things to check:


new firewall inbound rule inside the VM enabling sqlservr.exe. You can make it as restrictive as you want but start with allowing all inbound connections first to verify it works then lockdown one thing at a time 
new firewall inbound rule inside the VM enabling sqlbrowser.exe. Needed if running named instance. 
new endpoint for the Azure VM (via Azure portal) to enable SQL
Server port. Some users specify some arbitrary unused port for public
and SQL Server's default 1433 for private to reduce noise from port
scanners that roam the internet looking for common targets. There are pros/cons to this or using default for both. You can
decide what's appropriate or talk to your security folks for
guidance. 
TCP/IP enabled for SQL Server, should be enabled by default but takes 30s
to verify via Configuration Manager
SQL Server enabled for remote access, yes another obvious one but never hurts to verify