No Conservative government has served an entire term with an outright majority since Thatcher in the 1980s. Johnson is in a sense "unelected", but his premiership is no less legitimate than that of Cameron or Major.
Like it or not, Johnson was the "biggest" voice behind the successful Brexit campaign, and it is therefore appropriate that he is tasked with making Brexit happen, and taking responsibility for its success or failure.
Its completely fair to say that Brexit is an economically stupid and possibly xenophobic project, but its not fair to say that Brexit is a threat to democracy. Brexit is exactly what democracy looks like at its best: the people forcing the ruling classes to make decisions that are against the ruling class's own interests.
EDIT: Johnson is of course the head of government, not the head of state, since the UK is a monarchy.