In the Dragon Ball Super it is mentioned that only a God can sense other God(s). This being true, North Kai and Supreme Kai(s) were able to sense movements of Lord Beerus, but Goku wasn't.
Then how is it possible for Goku to sense North Kai and Supreme Kai(s)?
According to Dragon Ball Wikia, being a God (Divine Being/Non Mortal) and having Godly Ki are two different things. Goku and Vegeta are normal mortals but they still can harness God-Ki because of training with Whis.
Now on to your second question. As I explained above God-Ki can be obtained through training. Currently Goku and Vegeta has both the ki (Normal ki and God-ki). It is totally dependent on the characters which to use, if Goku decides to use only god-ki then his friends like Krillin would never be able to sense his power (ki). He uses both forms of ki. Even Supreme Kai have both ki, that's the reason Goku could sense him. As for proof you can visit Dragon Ball Wikia: http://dragonball.wikia.com/wiki/Ki#Deities.
It states:
Deities use a rare form of ki, known as God Ki (神の気, Kami no Ki), which is referenced as early as the Majin Buu Saga, but fully introduced in the movie Dragon Ball Z: Battle of Gods. It has been shown to be far above the normal usage of ki. Despite this type of energy dwelling specifically within gods and other high-level deities, god ki can be obtained by mortals via various means, including special rituals passed down through legends, and highly intense training under the supervision of an angel.
Godly ki is a more powerful version of ki. As seen during the God of Destruction Beerus Saga, Goku is unable to be on par with Beerus until he absorbs the power of Super Saiyan God. Goku is able to briefly fight on the same level after having lost his Super Saiyan God form when his mind was adjusted to Beerus's level of power and speed. However, it is implied to have been a temporary instance. After Goku and Vegeta train with Whis, they are able to use the god power in their base and Super Saiyan forms, making the transformations more powerful at will.
The God of Destruction Beerus, his mentor Whis, their siblings, and Saiyans who obtain the power of a god are able to use this form of ki, all of them vastly surpassing almost everyone else. The Supreme Kais also possess both normal ki and godly ki as they are "Gods of Creation", the opposite of Gods of Destruction, and both Kibito Kai and Old Kai are able to sense Beerus as well. It is implied that, since they possess both normal and godly ki, they lack the strength of those with purely godly ki. Kibito also appears to have godly ki, as Dabura states that neither his or the Eastern Supreme Kai's energy can be used to release Majin Buu from the Sealed Ball.[6]
Several beings who do not possess godly ki are capable of sensing it: Dende possesses the ability to sense godly ki due to the fact he is Guardian of Earth (which would imply that Kami and the other Guardians of Earth also possess the ability to sense it as well when they held the position) and is able to sense Beerus's ki.[7] After greatly powering up to the level of the gods, both Goku and Vegeta also possess the ability to sense godly ki. Objects like scouters are unable to scan the power level of a user of godly ki. In Dragon Ball Super, Frieza gains the ability to sense godly ki after his four months of training as he can sense Super Saiyan Blue Goku's ki and feels that it is less powerful than the normal ki of his Golden Frieza form, certain beings with power or strength that surpassed Super Saiyan God (such as Future Trunks' power increase after his transformation into Super Saiyan Rage, and Frieza's clan members' power increase after achieving their Golden forms) can sense Godly ki.
Due to Goku's body being taken over by the Supreme Kai apprentice of Universe 10, Zamasu, and then engraving Goku's fighting power into his own, Goku Black possess godly ki. Every time Black transforms into his Super Saiyan Rosé form he covers himself with Godly ki.
Saiyans are unable to sense the Ki of a God. Since King Kai doesn't qualify Goku is able to sense him.
What doesn't make sense to me is why Goku doesn't just become SSJ Blue and try sense the Ki of the Gods that he is aware of.
