If a Datagroup DATA_DG has been allocated 5GB and is full. How do we add and allocate more space to a DataGroup.
There is no such thing as a +"data group+" in Oracle.
ASM uses disks and diskgroups. Oracle uses tabespaces and datafiles (and also extents, segments and data blocks).
If a diskgroup is full, you can either create a new diskgroup as storage container, or you can add more disks to an existing diskgroup.
The adding disks to a diskgroup is not straightforward. Disks in a diskgroup is striped. Thus the new disk added, will become part of the striped set - and this requires what ASM calls a rebalance to occur, in order to make that new disk part of the stripe. If the disk is normal redundancy (mirrored) adding a single disk will only increase space for one part of the mirror. Another disk needs to be added to grow both mirror sets the same size.
5GB diskgroup is also questionable. Disks are typically 512GB and larger. Partitioning disks and creating diskgroups on partitions, is not recommended.
Make sure you have read the ASM documentation and understand how to use ASM to manage storage effectively and optimally.