Thermal relativity The more energy we add (heat) to a substance, say water, the higher Kinetic energy its particles have. Since the Kinetic energy is linked to the time taken by particles to move a certain distance and taking time dilation equation into account, the speed of particles will appear smaller to an observer, i.e he would find it cool instead of hot. Is this correct? 
 A: As a commenter stated, the speed of particles will depend on the observer.  If the bulk substance and the observer are in the same inertial frame, the moving particles' internal clocks will run slow relative to the observer, but the observer can correctly measure their speeds.
relativity and internal energy
The internal energy of a substance which is related to temperature comes from the motion of the particles that make up the substance.  As you add heat to the substance the particles will jiggle around faster.
If the particles are moving so fast that special relativistic effects start to matter, the simple $E_\mathrm{k}=\frac{1}{2} m v^2$ picture of kinetic energy is incorrect.  For non-interacting, monatomic particles in an ideal gas the relativistic energy of a particle is $E_\mathrm{tot}=\gamma m c^2$, where $\gamma$ is the Lorentz factor:
$$\gamma = \frac{1}{\sqrt{1-(v/c)^2}}$$
The kinetic energy of a particle would be:
$$ E_\mathrm{k} = E_\mathrm{tot} - E_\mathrm{mass} = (\gamma-1)m c^2$$
For real substances there could be interaction energies or other kinetic energies like rotational or internal molecular vibrations that also contribute to the total internal energy and change this a bit.
