It is generally not illegal for an individual to arbitrage between crypto-currency exchanges, and many people have done this without legal problems. However, the laws that apply to you will depend on which jurisdiction you live in, as do the tax liabilities you may incur.
Another risk you should be aware of is that if you're moving large amounts of money through a personal bank account it's possible that your bank may find your activity suspicious. Depending how helpful they are they may ask you to explain your transactions, or they may simply close your account.
Finally, be aware that there are already experienced people arbitraging between exchanges, so if there is a large, persistent gap between them that has not already been arbitraged away, there is often a reason for it. For example, at one point it was possible to earn money easily by buying bitcoins, selling them on Mt Gox, withdrawing Japanese Yen to a Japanese bank account, converting the Yen to USD, and using the USD to buy more bitcoins. Then one day it was no longer possible to withdraw the Japanese Yen.