The very opening of Act II scene 1 indicates the conflict between Alonso and his brother, Sebastian. Note the context. As the scene opens, Alonso is clearly struck massively by grief because he fears his son, Ferdinand, has died in the shipwreck. Gonzalo, the faithful retainer, tries to comfort his lord, but Sebastian only mocks his brother cruelly, given the situation. Consider what he says to Antonio as he listens to Gonzalo's words of comfort:
He receives comfort like cold porridge.
Such childish attempts to make fun of his brother and of Gonzalo show Sebastian to be a petty, vindictive individual who, as the scene develops, is shown to be actively opposed to his brother. He is very easily tempted into trying to kill Alonso by Antonio, even when Alonso seems to do nothing to warrant an emnity between himself and Sebastian. Even when Sebastian apparently tries to comfort Alonso, he ironicaly offers comfort that can only be described as "cold porridge," when he tries to say that Alonso should be glad for his loss as it means his daugher is far away from him and won't be able to sadden him with her presence. Hardly sympathetic words.