Reinforce master link
Sounds like your chain is not suitable for the master link, or the master link is not closing right.

There's no need to use a master link if you prefer to use the more traditional pins.  If you only ever take a chain off to change it, the advantages of a master link are not relevant.

Its possible to use sliding lockjaw pliers to push a pin in, but you run the risk of twisting it and making a poor seat.  Buy a chain tool if you're serious about riding a bike, or ask around other riders who may have one already.

If you can provide more info on your master link (like a clear photo) then it may be obvious to the rest of us what is happening.