How did Judaism spread to the kingdom of Israel?

By the time the kingdom of Israel was created by Saul, Judaism had existed there for over three centuries. As soon as the Israelites entered and took the Promised Land under the leadership of Joshua (Joshua ch.3-4), The Torah was immediately the religion of the land. It had been founded still earlier, by Abraham.

Judaism, more or less as we know it today, began with the religious reforms of King Josiah of Judah, the kingdom to the south of Israel. Undoubtedly, the Judahites had previously worshipped Yahweh as their chief God, as had the Israelites prior to the final destruction of Israel in 722 BCE, but this was as part of a polytheistic religion in which it is now known that the goddess Asherah was the consort of God.

Some historians have seen the Judahite king, Josiah as establishing his rule over the Assyrian province of Samaria, the former northern kingdom of Israel, at the time that Assyria was fighting for its life, though his rule beyond Judah cannot readily be established. In any case, Josiah did carry his reform measures into Bethel and the other cities of the former northern kingdom of Israel. A substantial proportion of the Samarians were descended from immigrants after the Assyrian dispersal of the Israelites, and even the ancestral Israelites had been polytheistic, so the introduction of Judaism could not have been easy. The coastal provinces, including Dor, were probably unaffected at this stage.

After the Jews returned from Babylonian Exile and re-established themselves in Judah, further efforts were made to spread Judaism to the former northern kingdom, but the people continued to worship their ancestral gods, while adding worship of the Jewish God.

By the Hellenistic period, a form of Judaism was widespread in the territory known several hundred years earlier as Israel.

As soon as the Israelites entered and took the entire Promised Land under the leadership of Joshua (Joshua ch.3-4), The Torah was immediately the religion of the land. Later, when the kingdom split into Judah and Israel, the Israelites continued in the original tradition/religion, though fewer of them were pious than in Judah.See also:

The Jewish religion was founded by Abraham

Jews have always worshiped the One God. God wrecked the Egyptian idols (Exodus 12:12) and warned against idolatry (Exodus 22:19). Abraham worshiped "the Lord God of Heaven and Earth" (Genesis 14:22 and 24:3) and complained about the Philistines' lack of fear of God (Genesis 20:11).

Jacob confiscated the idolatrous images taken from Shechem (Genesis 35:2) and got rid of them (Genesis 35:4); and refrained from invoking the gods of Nahor (Genesis 31:53). Rachel pilfered Laban's statue-images (Genesis 31:19) in order to prevent him from idolatry (Rashi commentary, ibid.). Joseph placed his hope in the God of the Forefathers (Genesis 50:24). Moses characterized the Golden Calf as "a great sin" (Exodus 32:21, 30) and punished the worshipers (Exodus ch.32). During the rest of his lifetime and that of Joshua (Judges 2:7), no incidents of Israelite idolatry were reported.

Shortly before he died, Moses warned the people that he suspected that they would eventually succumb to the lure of the idols (Deuteronomy 29:17). Joshua gave a similar warning (Joshua ch. 24).
These warnings came true. Many of the Israelites went astray after the foreign gods (Judges 2:11). However, they never invented their own idol. It was always the baneful influence of other peoples. And there were times when the entire Israelite nation repented (Judges 2:1-4) and prayed to God (Judges 3:9, 3:15, 6:6, 10:10). Those who did sin did not represent normative Judaism. They were deviating from the Torah's standard; they were publicly, repeatedly, and scathingly excoriated by the Prophets, and they caused God's retribution to come upon the entire people.

Because of the idol-worship that did happen, ancient images of idols have been found in Israel too. Images of God aren't found because it is forbidden to represent Him through imagery (Deuteronomy 4:15-16).

It should be noted that idolatry was never universal among the Israelites. The tradition of the One God was handed down in every generation, whether by the few or the many; and it is those who handed down that tradition whose beliefs we Jews continue today.

Deborah ascribed victory to God (Judges 4:14), Gideon tore down the idolatrous altar (Judges 6:25-27); Samson prayed to God (Judges 16:28), as did Hannah (1 Samuel 1:11) and Samuel (ibid. 12:18); Eli blessed in the name of God (1 Samuel 2:20), Saul built an altar to God (1 Samuel 14:35); Jonathan ascribed victory to God (1 Samuel 14:12), as did David (1 Samuel 17:46); and Solomon built the Temple for God (1 Kings 8:20). A number of the kings "did what was right in God's eyes": David (1 Kings 15:5), Solomon (see 1 Kings 3:3), Asa (1 Kings 15:11), Yehoshaphat (1 Kings 22:43), Yehu (2 Kings 10:30), Yehoash (2 Kings 12:3), Amatziah (2 Kings 14:3), Azariah (2 Kings 15:3), Yotam (2 Kings 15:34), Hizkiah (2 Kings 18:3), and Josiah (2 Kings 22:2). Even at the height of the unfortunate spread of idolatry among the less-loyal Ten Tribes, there were thousands who remained loyal to God (1 Kings 19:18).
And, of course, the Prophets, who spoke in the name of God and warned against idolatry: Isaiah, Jeremiah, Ezekiel, Hosea and so on.
The sages of the Talmud, who ridiculed idolatry (Megillah 25b), were simply continuing in the tradition of the Prophets whose verses are quoted in that context (ibid.).

See also:

Did Josiah change anything?

Jewish history timeline

Are Hebrews, Israelites and Jews the same people?