Describe the relationship between utilitarianism and self-interest.

Utilitarianism is nearly the antithesis of self-interest because one's own interests are to be given no more consideration than those of any other person. That is, decisions are to be impartial. The interests considered thus are not just those of the decision maker but those of everyone affected by the action.
Utilitarianism coincides with self-interest only when societal consequences are aligned with private consequences. One role of government is to establish institutions that provide this alignment.
When alignment has been achieved, individuals can evaluate actions in terms of the consequences for themselves rather than having to take into account the consequences for all others affected by an action.
Many of society's institutions are based on aligning the interests of firms and individuals with the aggregate interests of society. The institutions of private property and markets provide one means of alignment, since a voluntary transaction, such as a supply contract between two firms, makes both the buyer and seller better off. In the case of environmental protection, cap-and-trade systems align the interests of polluters and society.
The law of torts is an institution that assigns the social costs of accidents to individuals and firms, providing incentives to minimize the social costs of accidents and the cost of avoidance.
When markets are competitive and institutions are in place that align self-interest with societal well-being, the maximization of profit by a firm results in the greatest aggregate societal well-being; that is, the greatest difference between societal benefits and societal costs.