How fast would a spaceship have to go to reach Alpha Centauri within a person's lifetime?

If you are not interested in relativistic effects, the answer to your question is easy to workout. According to Wikipedia, Alpha Centauri is 4.24 ly away (4.0114x$10^{16}\mathrm{m}$). So to get there in 60 years ($1892160000\mathrm{s}$).

So your non-relativistic answer is

$v = \frac{d}{t} = \frac{4.0114 \times 10^{16}}{1892160000} = 21200000 \mathrm{m}\,\mathrm{s}^{-1}$.

This is 21200 $\mathrm{km}\,\mathrm{s}^{−1}$. The fastest recored space flight was 24,791Mph which is around 11$\mathrm{km}\,\mathrm{s}^{−1}$ which is 0.05% of 21200$\mathrm{km}\,\mathrm{s}^{−1}$. This means we have to be able to get spaceships to travel 2,000 times faster than the fastest current spaceship.

Note, I believe satellites in geostationary orbits do $\approx 17\mathrm{km}\,\mathrm{s}^{−1}$.

Edit. The relativistic calculation can be found here.

The distance between Earth and Alpha Centauri is $4.4\,\text{ly}$.

Dividing by $60\,\text{years}$ it's approximately $22000\,\text{km/s}$.

The relativistic factor, (I mean $\gamma = \frac{1}{\sqrt{1-v^2/c^2}}$) for this is almost $1$.

If we take a constant acceleration of $2g$ (it's possible) it would take only $320\,\text{hours}$ to reach this speed (and, same amount to stop). It total that's only $28\,\text{days}$. Negligibly small in compare with $60\,\text{years}$.