Is it annoying to carry a backpack when riding a road bike? I plan to buy a road bike for my daily commute but I need to carry my PC with me as a well-equipped engineer. My daily commute is around 5 km plus intermittent city tours. I afraid after some kilometres, it might make my back ache. Do you think it will be painful after a while? Do you prefer something like a trekking bike for this example?
Backpacks have a number of drawbacks when riding.  


They can make your back sweaty.
The can interfere with your helmet/head posture depending on bike geometry and backpack size/design.
They increase the weight on your hands.


In general most people I know that complain about riding with a backpack "feel" the extra weight as pressure on their hands, not in their back.  Obviously your bike geometry and anatomy may vary.

Backpacks also have some advantages.


Their weight can be moved dynamically by your torso.  The extra weight is on you and moves with you.
You have less to do at your destination.  The backpack is attached to you and doesn't need to be removed when you arrive.
You aren't required to alter your bike.
The backpack can be used for other things besides cycling.


My personal opinion was that for short commuting (30 minutes or less of cycling) I prefer a backpack.  It's less to deal with and a short enough time that the extra weight on my hands didn't matter to me.  For longer rides I would not wear a backpack because the extra weight becomes noticeable.  

Road bikes tend to have "longer" geometry and stretch you out.  This means you are holding your head up at a higher angle.  I would test your setup out before I committed to a road bicycle/helmet/backpack combo.  Many of the backpacks I have tried come up high enough that if you lean far enough forward, you can't actually raise your head all the way up with a helmet on, which isn't safe.