How can I kill an enemy under mind control?

No, you can't directly attack mind-controlled aliens without explosives. You can try and move out of range so that it will have to spend its turn closing in on you, giving you a free shot, or putting it into a disadvantageous position, such as being out in the open while surrounded by several of your operatives in high cover on overwatch). If there is fire on the map you may try to roast it alive, but it probably won't contirbute significant damage.

Note that a mind-controlled alien dying, explosives* or not, will result in its equipment being given to you at the end of mission, as if it was actually part of your squad. Also, higher-end enemies tend to have grenades or, in the case of Ethereals, an AOE psi attack, so you don't have to waste your own explosives killing them. For example, 3 perfectly healthy Elite Mutons under mind control can be made to kill themselves by having them huddle together and throw grenades at their own feet.

*In Enemy Within, explosives will destroy the equipment of mind-controlled aliens if they kill the alien (but not if they only wound it).

You can use area of effect (AoE) attacks like grenades, rockets, Mind Rift (Ethereal ability), make the mind controlled enemy step into Poison Cloud or be poisoned by Chryssalid's bite. If you can't make anything of listed above, you can make this enemy unit skip his turn when the Mind Control wears off. To do so, in the last turn, just before you lose control of the enemy, place at least one of your warriors nearby this enemy, but behind the cover (partial or full - does not seem to matter). Make this warrior Overwatch. Enemy will try using his special ability which directly harms warrior (e.g. Thin Man will try to poison your warrior, Muton and Muton Elite will try using grenade, Ethereal will use Mind Control, Mind Blast or even Mind Rift).

This is in details described on XCom Wiki.