Recently, I got a new pc, an AMD A6-6400K APU with Radeon(tm) HD Graphics 3.90 GHz. According to the page for the Eternal Embers DLC on Can You Run It, it meets the requirements for recommended requirements for the best performance. Yet, the game doesn't run very smoothly. It suffers from poor frames per second. According to the performance tab in the Task Manager of my pc, it only has one core.

Why am I getting such poor performance? What can I do to have a much better optimal and smooth gaming experience?

Excuse me but this "PC" is a Laptop right?
The CPU is about 10 Years old. What do you expect from
such old Hardware? It only has 2 cores...
The HD Graphics is not suitable for playing games.
You really need a PC for playing games.

I guess the requirements on the store pages are still from the original release of Titan Quest baxk in 2006 and never got updated. The newer DLCs are more performance hungry (have more objects in the environment) than the original levels, so that may add to your struggles.
The engine is single-threaded, and it may help to use task manager to set affinity to the second core - removing it from the first core seems to make it run more smoothly.
With your graphics that old, you may be better off with the DirectX 9 renderer (DX11 is the default).
In graphics settings, I'd try first to turn off detail objects - that sounds like it should reduce the load in the newer DLCs.
Also shadow quality and reflection quality tend to have a large performance impact on graphics.

I got a new computer a week ago, an intel core i7. There is an incredible difference in performance on this one from the touchscreen.