Bicycle dull noise when pedalling
My first thought was that you are getting some rubbing of the chain on the front derailleur. Rubbing would only occur when in highest or lowest gears though, and you don't mention that. You can check for this by holding the rear wheel off the ground, pedaling manually and inspecting the chain where it passes through the front derailleur, if you want to rule this out.

Unfortunately I think that a problem with the bottom bracket bearings (bearings that the crank axle runs through) is more likely, especially as you say you can feel vibration through you feet. To check for this manually turn crank arms on either side of the bike, you may feel some resistance, roughness or clicking as the crank turns. (If you can get the chain out of the derailleur cage and off the chainrings it makes it easier to feel this.) You can also try to rock the crank in the frame, there should be hardly any play between the crank and the frame. 

From the video I can see you have a cartridge type bottom bracket. The cranks on either side are attached via a square or splined taper on the end of the axle. These cannot be adjusted or repaired and simply have to be replaced. They are not particularly expensive or difficult to install, but do require some special tools (a crank puller, BB removal/installation tool and if you're doing things right, a torque wrench). 

This video shows what's involved and the tools needed.