How to restore my Local database to Windows Azure Database?

There are couple of ways to do that:

1. The most reliable (that I usually use) is using the SQL Azure Migration Wizard
2. Then you have the "Generate Scripts" option in your SSMS (SQL Server Management Studio 2008 R2 or later). The generate Scripts option does have a feature to make SQL Azure compliant script. It can also script data. But I suggest using method one if you want to migrate data, too. SQL Azure Migration Wizard uses BCP tool, which dumps the data into binary files, while SSMS just generates inline INSERT Statements
3. SSIS (SQL Server Integration Services) - you can create a Data Transfer task, but I would use it very rare for small DBs.
4. Export/Import data tier application (using DACPAC package) - this will migrate your schema and can migrate your data as well.

UPDATE 2017-10-24

Azure SQL Database now supports transactional replication from your on-premises SQL Server. You can configure your Azure SQL Database as subscriber and your SQL Server on-premise as publisher. You can read more about database migration options and strategies here: https://docs.microsoft.com/en-us/azure/sql-database/sql-database-cloud-migrate

Update 2017-12-23

SQL Azure Migration Wizard has been replaced by Data Migration Assistant: https://docs.microsoft.com/en-us/azure/sql-database/sql-database-cloud-migrate

If you already have a database on local machine then you can migrate it directly on Azure using SQL Management Studio. Follow these steps:

1. Right click on the database name in SSMS

2. Follow the wizard that appears then

There are a few ways in which to achieve this functionality.

1. SQL Azure Import/Export. "DACPAC" & "BACPAC"
2. Microsoft Codename "Data Transfer"
3. SQL Azure Data Sync Windows Azure SQL Data Sync
4. Migrating Databases using Import/Export Wizard in SSMS
5. SQL Azure Migration Wizard (As Mentioned in other Answers)