How to Take a Time Machine Backup on a HDD which already has important data?

The recommended way is to create a new partition dedicated to Time Machine.

Creating a partition isn't hard. Restart with command - R held down and open Disk Utility.

Step 1 - select the top level hard drive.

Step 2 - select the Partition tab.

Step 3 - grab the //// to slide the partition up to make room for a new partition.

Step 4 - select the vacant space and hit the + sign.

Step 5- Mac OS Extended (Journaled) and add a name.

Step 5a, click and select GUID.

Then Apply.