Can you please suggest a way so that I could partition few existing tables of my Azure SQL Database.
User-defined filegroups are not a requirement for table partitioning. With Azure SQL Database, just specify the PRIMARY for all filegroups in the partition scheme.
Below is a basic T-SQL example to create a new partition function, partition scheme, and partition an existing table. The DDL for your specific need will vary depending on the indexes on the existing tables and whether you want to partition those as well (align).
--create function with yearly boundaries
CREATE PARTITION FUNCTION PF_DatetimeExample (datetime)
AS RANGE RIGHT FOR VALUES('20180101','20190101','20200101','20210101');
--create scheme for the function
CREATE PARTITION SCHEME PS_DatetimeExample
AS PARTITION PF_TEST ALL TO ([PRIMARY]);
--partition the table
CREATE CLUSTERED INDEX cdx ON dbo.YourTable(YourDatetimeColumn)
ON PS_DatetimeExample(YourDatetimeColumn);