What were the complaints against the king of England in the Declaration of Independence?

In answering this question, it's important to understand the importance of the Declaration of Independence's overall rhetorical strategy. It was written primarily to persuade Americans who might otherwise be skeptical at the very idea of independence—that this was the only way forward. It was also written with the intent of persuading the rest of the world that the American colonists had been unfairly treated at the hands of their colonial overlords and that this why they had decided to bite the bullet and declare independence from Great Britain.
In general terms, the Declaration expresses the colonists' conviction that they'd been pushed into a corner by the arrogance and intransigence of the British and therefore had no alternative but to declare independence:

Such has been the patient sufferance of these Colonies; and such is now the necessity which constrains them to alter their former Systems of Government. The history of the present King of Great Britain is a history of repeated injuries and usurpations, all having in direct object the establishment of an absolute Tyranny over these States.

In other words, they didn't want to have to do this, but they argued that they were forced into it by the British. It was the British arrogance, intransigence, and unwillingness to listen to their perfectly reasonable complaints that led to the crisis. In riding roughshod over the ancient liberties of the American people, they believed that King George III and his ministers had sought to establish tyrannical rule over the colonies.
Strong words indeed. And although there's more than an element of hyperbole involved here, there can be no doubt that this particular passage of the Declaration is an accurate reflection of public opinion in the American colonies at this time.

Anywhere a reader spots the word "he" in the Declaration of Independence, the pronoun reference is to Britain's King George III.
The list of abuses of the King begins with

He has refused his Assent to Laws, the most wholesome and necessary for the public good

and continues through

He has excited domestic insurrections amongst us, and has endeavoured to bring on the inhabitants of our frontiers, the merciless Indian Savages, whose known rule of warfare, is an undistinguished destruction of all ages, sexes and conditions,

ultimately listing twenty-six separate complaints in George III's treatment of the colonies.
To paraphrase some examples: the King has prohibited the passing of laws that would help the colonies and prohibited the royal governors in the colonies from passing laws without his oversight even when there is a time-sensitive need for it. He won't pass laws benefiting large swaths of the colonies unless they give up their right to representation in Parliament. He uses tactics like calling meetings that are hard for colonists to attend, knowing that his emissaries can act in their absence. He dissolves local legislatures at will and doesn't move to replace them even when laws are needed in those areas. He is controlling the population of the colonies by preventing emigration. He won't empower independent courts in the colonies and handpicks only judges who will do his bidding. He has expanded bureaucracy in the colonies. He orders military occupation without cause and enacts martial law.
There are fourteen more complaints that can be paraphrased. The examples above are a good representative example of the tyrannical rule that the king held over the colonies before the Revolutionary War ended it.

Here are some complaints about the king in the Declaration of Independence:

He has refused his Assent to Laws, the most wholesome and necessary for the public good.

Translation: The king has refused to pass certain laws that would benefit the citizens of the colonies.

He has forbidden his Governors to pass Laws of immediate and pressing importance, unless suspended in their operation till his Assent should be obtained; and when so suspended, he has utterly neglected to attend to them.

Translation: The Governors of the Colonies cannot pass emergency laws to address emergency situations, because all laws must go through the king himself.

He has called together legislative bodies at places unusual, uncomfortable, and distant from the depository of their Public Records, for the sole purpose of fatiguing them into compliance with his measures.

Translation: He is making colonial legislatures travel great distances to spitefully make extra work for the lawmakers and to pressure them into giving him what he wanted in the first place.

He has obstructed the Administration of Justice by refusing his Assent to Laws for establishing Judiciary Powers.

Translation: He has refused to allow the colonies to develop their own court systems for enforcement of their own laws.

He has kept among us, in times of peace, Standing Armies without the Consent of our legislatures.
He has affected to render the Military independent of and superior to the Civil Power.
For quartering large bodies of armed troops among us.

Translation: Without our agreement, he has created large armies, prevented us from controlling those armies, and forced us to find housing for those armies.

For imposing Taxes on us without our Consent:

This one needs no translation, but it's a very important one!

He has excited domestic insurrections amongst us, and has endeavoured to bring on the inhabitants of our frontiers, the merciless Indian Savages whose known rule of warfare, is an undistinguished destruction of all ages, sexes and conditions.

Translation: He has tried to rile up our enemies on the continent, including our own rebels and the Native American tribes.
There are quite a few more; take a look at the text itself for the long list.
https://www.ushistory.org/declaration/document/