The connection to the tribal offerings to Chanukah is in the use of the term Chanukat Hamiuzbe'ach as seen in Bemidbar 7:11."Chanukah" means "dedication", and on the actual Chanukah, in a sense, the Bet Hamikdash was re-dedicated.
With some variations, the reading each day corresponds to the numerical day of dedication as outlined in Bemidbar 7:12-73.
On the 1st day, by way of introduction, verses 7:1-11 are read for Cohen. Levi and Yisra'el split the reading for the 1st day of dedication 7:12-17.
Each day following this, Cohen & Levi split the reading for that day (2nd day Hanukkah starts with reading of 2nd day dedication) and Yisra'el gets reading for the following day.
On Shabbat Chanukah, maftir is the reading of the dedication day corresponding to that day of Chanukah. So 4th day Chanukah = 4th day of dedication, etc.
The 6th day of Chanukah is always Rosh Hodesh. On a weekday, the first 3 aliyot are the Rosh Hodesh reading, and the 4th is from the 6th day of Chanukah.
If the 7th day Chanukah is also Rosh Hodesh, same idea as above with 4th reading for 7th day Chanukah.
8th day Chanukah - Cohen and Levi split reading for 8th day and Yisra'el completes the entire list from 9th day through 12th day and continues the reading until Benidbar 8:4 which ends with a discussion of lighting the menorah.
On those occasions where Shabbat, Chanukha nd Rosh Hodesh all coincide, 3 Torot are used- First 6 aliyot from Miketz, 7th from Rosh Hodesh and maftir for 6th day Chanukah.