Can I add an exterior outlet by tapping an adjacent interior outlet?
Generally speaking, this is OK. Bedroom receptacles do not need to be on dedicated circuits, so adding another receptacle to the circuit is not, in and of itself, a problem. But there are a few things to watch out for:
15A vs. 20A?
Your bedroom receptacle might be on a 15A circuit or a 20A circuit. If the electric grill only needs a 15A circuit then there is no problem. However, if it needs a 20A circuit and you only have a 15A circuit available then you need to replace the breaker (usually very easy) but you also need to make sure that all wiring on the circuit is 12 AWG or larger, which is unlikely because if it was then there would have been no reason NOT to have had a 20A breaker in the first place. If any of the wiring is 14 AWG then you can't upgrade to 20A without replacing the wiring. If the circuit has even a few hundred watts of constant use - e.g., a desktop computer - then you may find that the circuit does not have enough capacity to run the electric grill and the usual loads at the same time.
Don't remove the receptacle
You may be tempted to remove the existing receptacle and simply splice the wires in the junction box using wire nuts. Most likely you can't do that because of requirements for, generally speaking, receptacles along every wall. So either pigtail to add new wires for the outside receptacle, or use the second set of screws on the receptacle if not already in use.
GFCI
Bedroom receptacles typically require AFCI (though if it is an older house then they will likely not have AFCI) but do not normally require GFCI protection. Outside receptacles require GFCI. You can replace the breaker with a AFCI/GFCI breaker, replace the bedroom receptacle with a GFCI/receptacle and put the new receptacle wires on the LOAD screws, or install a GFCI/receptacle outside. Installing GFCI outside is the simplest solution, but subjects the GFCI to weather - hot/cold/wet - which shortens the life of the GFCI electronics.
Weather Resistance
An outdoor receptacle should be weather resistant and have an in-use cover.