How can both kinetic and potential energy increase simultaneously in a open substance?
You are adding energy by heating, this generally does not break chemical bonds (unless the temperature reaches hundreds of degrees) but instead the weak intermolecular potential energy between molecules (sometimes call van-der Waals interaction) and so the solid will melt, then the liquid will boil, obviously :) 

The kinetic and potential energy of the sample has increased. Kinetic energy in translational motion of the molecules and internal potential and kinetic energy of the molecules has also increased as they have absorbed some of the heat as vibrational and rotational energy.  (A non-linear molecule with N atoms has $3N-6$ different ways of vibrating, e.g ethane has 18 ways). 

The reason that the temperature increases rapidly is due to the fact that if you heat a pan on the stove, or a boiling tube with a Bunsen burner  is that convection of heat away into the air is very poor. So your boiling tube is effectively insulated. If it were completely insulated then less added heat would be needed to reach the same temperature. As an admittedly odd example, you will know that a plumber can join Cu pipes with solder using a blow torch, but only if they are dry. If they contain a even small amount of water more of the heat is carried away (and heats the water) and soldering becomes impossible.

If you heat a sealed tube, it is likely to explode violently due to increase in pressure; so don't do it! (To prevent explosion is why pressure cookers have a pressure release value).