Driving a 6V relay from an 3.3v GIO pin of Raspberry PI3
The item you linked in your question is a logic level converter and so this would probably be unsuitable as it wouldn't be designed to work with currents around 100 mA. 

The simplest implementation I can think of is this: -



It shows a 5 volt supply and relay but this will work with a 6 volt supply and relay. D1 is there to "catch" back emfs from the relay coil when you turn it off (MCU out = 0).

Choosing the base resistor at 1 kohm ensures that the transistor properly turns on when the collector load is only 49.5 ohms (55 ohm - 10%). The 2N2222 has a current gain guaranteed to be 100 at a collector current of 150 mA hence, with about 2.5 mA flowing into the base, the collector could deliver 260 mA should the load be much lower resistance. In this configuration I would expect the transistor to fully turn on to less than 0.5 volts and properly operate the relay. 

Short story: You need to drive the BJT's base with enough current and choose a transistor that is rated for a decent hFE at currents in excess of 100 mA.