W. L. Warren hypothesizes that Henry was most deeply motivated, especially in his earliest years, by wanting to restore the situation to what it had been in the life of Henry I his grandfather (regardless of how the situation may have changed). So he was driven about retaking absolutely everything that had been lost in the Anarchy, the Stephen/Matilda civil war (and this dogmatism got him into trouble with the Church and various lords).
But he was also driven by not taking what was not rightfully his. After the great rebellion, he had the king of Scotland's homage, but it was a loose and undefined overlordship, and there was some vague historical precedent that he could point to. Similarly with the overlordship of Welsh princes and in Ireland (though for Ireland, there was no precedent). Against the rebels, he restored them all to what they had before the war ... except that he took their castles and destroyed them, but it had been an old precedent that all castles were franchises from the king or owned by the king (I'm not sure of the exact notion), so even that was his by right. Despite being much richer and more secure, he took the Vexin (he claimed it as being a part of Normandy) but not south of it. He could have had early alliances with the counts of Champaign and Blois, cupping the Île-de-France (and in position to crush it like a nutcracker), but let them go. He let his sons, whom he intended to have Continental lordships, do homage to the king of France for their various lands.
The relationship between Henry II and the Capetian kings of France was complicated, and marked by greater strength on the part of Louis VII than has often been assumed in English historiography.
For one thing, Henry II's ancestors, the counts of Anjou had been long time supporters of the French monarchy; Jean Dunbabin argues that Louis would have seen Henry as a powerful, if dangerous, ally rather than a rival at the outset. This was the role that Henry played in 1158, for example. There were worrying signs; Louis was likely uncomfortable with Henry II becoming duke of Normandy, recognising his hereditary rights and upsetting Louis' plans to install his brother-in-law as duke. Still, at the outset Louis VII is unlikely to have been too concerned about Henry.
Louis realised early on the Henry's military resources, as duke of Normandy, outstripped his. For this reason he turned to political means to check Henry's ambitions.
Once Henry became duke of Aquitaine Louis moved to contain him politically in the south; he allied himself with the count of Toulouse and extended his patronage throughout the Languedoc, particularly to the church. He also secured a marriage alliance with Castille.
The key break was when Henry became king of England in 1154. Up until this point, things were progressing fairly well for Louis - yes he had been defeated militarily, but Henry had also been acknowledged Louis' over-lordship. From here on, however, Henry's power and resources, in material terms, outstripped Louis' - he was in a position to simply project his power over Normandy, as Henry I had done, regardless of what Louis thought. A clear elucidation of this was in the case of the powerful counts of Flanders - despite initial hostility from the counts Henry was able to simply buy them off. However even this didn't mean that Louis was reduced to a state of powerlessness.
Louis' response to Henry's overwhelming material superiority was to try and claim the moral and political high ground. A good example of this was the Council of Soissons in 1155. At Soissons Louis attempted to build a political coalition on the basis of a shared desire for peace, alliances and the over-lordship of the king of France - at was here that the principle that, if denied justice by their lord, a vassal could appeal to the king of France was established, for example. Dunbabin compares this to the imperial peaces of the Holy Roman Empire and argues that this was the first time the kings of France attempted to assert imperium.
Louis tried to involve Henry himself in a similar arrangement in 1158, an attempt which actually saw some success. Henry pledged fidelity to Louis, a pledge with forced him to back down from a campaign against the count of Toulouse when Louis' intervened in person. This policy of maintaining the moral high ground was a cornerstone of Louis' attempts to contain Henry. That said, Louis' power in the south wouldn't last and by the time of his death in 1189 Angevin power was ascendant in Toulouse.
It would be a mistake to think of the relationship between Henry and Louis as one of permanent acrimony. There were periods of warmth, the most prominent due to the betrothal Henry's son, the future Young King and Louis' daughter Margaret in 1160. Part of this arrangement was that Young Henry did homage for Normandy and the Vexin was made part of Margaret's dowry. This peace arrangement was subverted by Henry however, as he immediately received Papal dispensation for the marriage, despite the children's age, allowing him to immediately take control of the Vexin. This was the key breakdown in relations that led to a permanent state of acrimony between the two kings.
There were geographic flash-points for tension all over France. The two kings struggled over Berry inconclusively. Henry II was ultimately forced concede hegemony over Auvergne on his death bed. These areas, however, were not as critical as Toulouse and the Vexin.
As mentioned previously, a cornerstone of Louis VII's strategy in dealing with Henry II was to maintain the moral high ground and use it to seize political advantage. This had been a cornerstone of Louis' policy throughout his entire reign - he was deeply concerned with his standing amongst the princes of Europe. In this he was very successful, for example staying close to the Papacy of Alexander III, and employing his title of 'most Christian king' to maximum affect. This church-friendly stance found an obvious outlet in the Becket affair. Thomas Becket was a former close friend and adviser of Henry II whom he had appointed archbishop of Canterbury. However, once installed in the bishopric Becket transformed himself into a fierce advocate for the church, causing a rift with Henry that turned into an enormous political wound. Louis was able to exploit this by (after some delay) offering sanctuary to Becket and using the opportunity to encourage rebellion against Henry (which Henry argued was a breach of Louis' duties as lord). Throughout all of this, Louis maintained the moral high ground by positioning himself as champion of the church.
Louis and later his son, Phillip II's major source of success against Henry was exploiting Angevin family tensions. Henry's sons were all ambitious and felt stymied by their father. Louis and Phillip were able to exploit their position as king of France and therefore overlord of the French Angevin lands to present themselves in a paternal light, defending the the rights of their subjects, Henry's sons, that had been trampled by the nefarious Henry. This was the policy that led to Henry's major defeat a short time before his death, at the hands of Philip II and the future Richard I. This policy allowed the French kings to more successfully stymie Henry in the later years of his reign than they had earlier on.
In terms of Louis specifically, he built up both the prestige and finances of the French crown extensively, and both of these factors would be leveraged by his more militarily capable, energetic son Philip to slowly pick apart the Angevin Dominion over the course of the next few decades.
Hopefully what this post demonstrates is that the political situation in the later 12th century wasn't 'mighty Henry kicks around feeble Louis'; it was much more complicated. Henry was undoubtedly in the stronger position, but it wasn't a totally one sided affair. Henry certainly came out the better much of the time, but to reduce Louis VII to a mere punching bag is reductive. It's also worth pointing out that, whilst the acceptance of French over-lordship is often pointed to as an Achilles heel for the Angevins the reality of the situtation was more complicated. As overlord of Henry's French lands Louis was under intense pressure to recognise his rights, as well as play the part of good lord as much as Henry the loyal vassal. As John Gillingham has argued, the acknowledged over-lordship of the Angevin possessions in France is a far more complicated affair than is often assumed.
The point made be /u/scarlet_sage is a good one - Henry had no right to the French throne. Therefore, why would we expect him to try and claim it? That he controlled X percentage of the modern nation of French, and was therefore in a position to claim the throne is borne more out of Crusader Kings than 12th century politics.