retsuseiba: Sure, the 7750 is considered to be very energy efficient and some 7750 owners have said that a 300W PSU worked for them, however the AMD website recommends something around 400-500W PSUs for that card.
It's important to be aware of the fact that PSUs lose efficiency over time. It may output 300W now, but in a couple of years, you may be closer to, say, 250W.
From what I've been able to dig up, the peak load of the 7750 will be
55W, or 75W when using PowerTune to overclock. That should be low enough to use a 300W power supply, but depending on what else you have hooked up, you may have problems eventually.
There's also the question of amperage. Your power supply is 300W, so as long as your components don't draw more power than that, you should be okay, right? Not necessarily. Different components get power from different cables and at different voltages, and the power supply will only deliver so much power over a cable. GPUs use 12V, so if we go by the stated absolute max of 75W, that means the PSU has to be able to deliver 6.25A on that cable (since Wattage = Voltage * Amperage). A 300W PSU should have no problems with that, but the GPU is not the only device using 12V.
For a small PSU like this, it seems likely that it only has a single
rail. Rails divide the available power to the different cables, and each rail will only provide a certain amount of power. If the PSU has 2 12V outputs, each may be rated at e.g. 15A, but if you do the math, it clearly isn't possible to fully power both of them with only 300W. Each output is associated with a rail, so if it's a single-rail PSU, then that PSU would only have 15A for the two 12V outputs
combined.
For all of the reasons above, it can be a good idea to go for a bigger PSU. You don't want to make it too big, because a large PSU will generally be inefficient if too little power is drawn from it, but it's a good idea to have a buffer, especially if you plan on keeping it for several years. 400W is likely plenty here, but you may want to go for 450 or maybe even 500 if you replace the PSU, just in case you want to upgrade it further at some point.
But yeah, consider getting a computer which already has the right parts from the beginning instead (or buy parts and build your own; it's really not very hard).