Usually you cannot. A paperclip is soft, round wire with no hook or teeth, so it tends to slip off the key or push it further into the lock. If part of the key sticks out, fine pliers are a better bet; if not, leave it for a locksmith with an extractor.
The paperclip idea is to use bent wire as a hook and catch the key by its cuts. In practice the wire bends before the key moves. It is also usually thicker than the gap beside the broken piece, so it tends to press the piece deeper instead of catching it. Hair grips and safety pins have the same problem.
Deeper is the real risk. A key that has broken just inside the face of the lock is often a quick job for a locksmith. Once it has been pushed further in, it takes longer. The Master Locksmiths Association warns that home attempts can damage the inside of the lock badly enough to need a replacement.
What sometimes works on your own lock: if a clear piece of metal sticks out, grip it with fine needle nose pliers and pull straight back without twisting. If it slips or will not move, stop there. Do not use superglue, do not try to turn the lock with the piece inside, and do not push the spare key in after it.
If the piece is out of reach, we remove snapped keys for £85 by day, £120 in the evening and at weekends and £150 at night. Prices include travel and up to one hour on site, and often the key comes out without replacing the lock. Keep both halves of the key, because a new one can often be cut from them.
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