The Children of Israel wanted a king only to be like the other nations.
The sin – This approach says that the king the Children of Israel asked for was supposed to be like the nations and not like a king from the Children of Israel. The way the nation asked for the king was bad and not having the king itself.
"Like all the nations" – These are the words that are used to support this approach that the Children of Israel are focusing on that it is like the nations.
Verses in Devarim – The Abarbanel asks on this approach from the verses in Devarim were the commandment is that when they go in to the land and want a king "like all the nation" they are allowed to appoint a king. From these verses it sounds like that it is totally fine to want a king like all the other nations. It is possible to answer that Devarim is actually rejecting a king like that and that is why it commands after to get a king especially not from the Non-Jews.
Hashem's speech to Shemuel – The Abarbanel raises another question. When Shemuel and Hashem say that is bad neither one mentions that it is bad that they wanted a king like the nations. Rather just that there was a problem with having a king that would judge them or a king that would reject Hashem. If there was a real problem having a king like that it would seem that Shemuel and Hashem would talk about that.
Appointing a king at the end – Maybe Hashem does appoint a king for them but not one that is like all the nations. Instead like in Devarim Hashem appoints a king that is Jewish specifically and not an outsider.
The Children of Israel do not need a king at this time because Hashem is fighting for them.
The sin – Rabbi Yehuda in the Sifre does not give details to why this was a wrong time though the Ramban and Malbim explain. They say that at this time Hashem was fighting for the Children of Israel and he was doing to them miracles. When Hashem is doing all of the work you do not need a king but after when Hashem does not interact directly then one needs a king that can fight all the other nations.
"Like all the nations" – This approach presumably explains the verse that everybody else has a king that fights for them since they do not believe in Hashem but for the Children of Israel they did not need a king since they do believe in Hashem so they should not be like the nations.
Verses in Devarim – It is unclear how to explain the verses in Devarim. Although it is possible to say that when the verse says that after one conquers the land and what settles down in it one can ask for a king, it is referring after the time of Shemuel when one really does need a king, but it is not obvious.
Real sin – The verses do not raise at all that this is a wrong time for asking a king. It is unclear where they get this answer from.
Hashem's speech to Shemuel – Hashem's speech to Shemuel works well with this approach. According to this, the nation is rejecting Hashem because they are asking for a king when Hashem is still directly fighting for them as oppose to after.
Appointing a king at the end – The Ramban in his commentary explains that from the beginning Hashem decided not to give the kingship to Shaul since it was the wrong time. According to him the real kingship only started when it was suppose to after Shaul but Hashem gave a non staying kingship from the wrong tribe to Shaul.