Reading Transistor and Diode Spec
The 2N4401 will do fine in your circuit.  However, I would ditch the capacitor accross the relay.  Just about any regular diode will do, but the one you have is inappropriate.  That is a zener diode meant to act as a voltage reference.  You want a ordinary silicon or possibly Schottky diode.

To calculate the minimum specs required, you first have to identify the relay you want to use.  The relay datasheet will tell you the current it requires at 12 V.  That tells you directly the current the transistor must be able to handle.  Most small 12 V relays require 15-50 mA, which is well within the capability of a 2N4401.

You also have to think about how much collector current the transistor can support for the base current you are giving it.  Your sketch shows a 1 k&Omega; base resistor.  Figure the B-E drop as 700 mV, which leave 4.3 V accross the resistor when the relay is supposed to be on.  That means there will be 4.3 mA into the base.  Figure a 2N4001 can be relied on to have a gain of at last 50.  Your specs say 100-300 for Hfe, which is another way to say gain, but what you post is a dumbed down snapshot of the datasheet.  In any case, 4.3 mA times 50 is 215 mA, which is lots more than any reasonable "small" 12 V relay is going to require, so all is fine there.

The diode has to be able to take up to the relay current in forward mode, and block at least the power voltage in reverse mode.  A 1N4148 is a common small signal diode that can do this.  Those usually top out at around 50-75 mA (there are a lot of variants out there), but again, that is still more than a reasonable relay will require.