As a follower, the transistor will never actually switch on fully because of the Vbe drop. The PICAXE drives the base to +5v (or whatever the +ve rail voltage is at the time), and the emitter is about 0.7v below that. So there can never be less than 0.7v between the collector and the emitter. The power lost to heat because of this drop, for a current of 250mA, is about 175mW (Vce*Ic, in this case 0.7*0.250)
‘Ptot’ (Total power dissipation) is another factor that may need taking into account sometimes. But usually only when the transistor is used in its linear mode*. It is quoted in Watts and again should be a multiple of your anticipated power dissipation to leave a good overhead. The power dissipated by the transistor is the emitter current (Ic) * the collector emitter voltage (Vce). When the transistor is used as a switch and will normally have either a small Vce (when on) or a small Ic (when off), the dissipation is usually so low as to not be a consideration. (BTW, this is why PWM or ‘switching’ is so much more efficient than linear methods. Very little power is lost as heat in the transistor). When your transistor gets too hot, or the magic smoke is released, it usually means that you have exceeded the transistors rated Ptot.
*Linear mode means when Vce varies in an analogue manner (as in an amplifier), so that both Ic and Vbe are simultaneously significant. Since, in your circuit, the Vce never gets down to Vce(sat) (usually around 0.2v), the transistor is never saturated and is actually working in linear mode, though only just.
NOTE. When I said that I would choose the BC639, it was just because I happen to have lots handy (they are cheap) and it will do the job. But when my prototype was working, I would be tempted to get a better transistor for the finished product. Something newer, with a little more overhead on the stats.
Actually, for this application, I would definitely use a MOSFET with the source to 0v and the LEDs between +v and the drain. MOSFETs are excellent in switching apps like this, and draw almost zero current from the PICAXE, and would dissipate less power as heat.