The idea of God within can either be explained rightly, or really badly.
In the right understanding, Grace is the Holy Spirit entering us and infusing Divinity within us, making us a temple of the Spirit. Santifying Grace is partaking in Divinity, and can be understood as God placing Himself within us, to makes us Holy, for only God is Holy, so we must become God (by participation) in order to be Holy.
Moreover, I will give you a new heart and put a new spirit within you; and I will remove the heart of stone from your flesh and give you a heart of flesh. And I will put my Spirit in you and move you to follow my decrees and be careful to keep my laws. (Ezekiel 36: 26-27)
St. Catherine of Genoa also wrote a book on purgatory, in which she explains it as an inner fire which purifies us from within, which could be understood as santifying Grace, after being infused in our being, purifying us.
The difference between the Christian view and the Hindu view is that Hindus believe that humans are inherently Divine, and have just "forgotten" it. Christians, on the other hand, believe that humans are only Divine by Grace, a free gift given and not earned nor inherent in our nature, and primarily received in Baptism (where we are born again). In other words, Hindus reject the Creator/Creation divide, and advocates panentheism (in the mainstream understanding: there are many paths or yogas in Hinduism, many which lead to contradictory truths), while Christians are "quasi-panentheists."
To a Christian, only the Father (who is the source of Divintiy), the Son (who receives Divintiy through eternal begotteness), and the Holy Spirit (who receives Divinity through eternal "spiriting") are essentially Divine, and everyone else is Divine only accidently, only by Grace (we receive Divinity by participation/Grace/Divine Adoption). Hinduism claims everyone and everything is Divine essentially.
I find Hinduism to be one of the few religions that can actually rival Christianity, personally, in part due to how close its understanding is to the Christian understanding: a mixture of 90%Truth and 10%Lies is far more convincing than one of 10%Truth and 90%Lies.
Everything I wrote is based on multiple Fathers, St. Thomas Aquinas, and Father Reginald Garrigou-Lagrange. Here's a great quote from Father Garrigou-Lagrange (if you understand basic Aristotelian and Thomist philosophy and Western theological language):
On the other hand, sanctifying grace as such is not a participation in being in general, nor in life in general, nor in intellectuality in general, but a participation in Deity, which is found naturally only in God. Thus only grace is called a participation in the divine nature according as it is in us the radical principle of operations strictly divine, of which the formal object is (in heaven, at least) absolutely the same as the formal object of the uncreated operations of God.
All of this may be diagrammed as follows:
Thus the stone participates in being and has a likeness to God on the basis of being; grace, on the contrary, is directly and immediately a participation in the divine nature, not in any perfection analogically shared by God and the creature.
Therefore Deity as such cannot be partaken of except by some essentially supernatural gift. And, conversely, grace cannot be essentially supernatural unless it is a formal and physical participation in the divine nature as divine, that is, in the intimate life of God, or Deity as Deity, ordaining us to the knowledge of God as He Himself knows Himself immediately and to the love of God as He loves Himself.
Furthermore, sanctifying grace is a participation in Deity as it is in itself and not merely as it is known to us. For it is produced in our soul by an immediate infusion altogether independently of our knowledge of the Deity; and just as Deity as such is communicated to the Son by eternal generation, so Deity as such is partaken of by the just, especially by the blessed, through divine adoption.7
Hence, materially, grace is a finite accident, an entitative habit, but formally it is a formal participation in Deity as it is in itself, as it subsists in the three persons. Thus it is clearly evident that Deity as such in a certain sense surpasses being and intellection, since all absolutely simple perfections are identified in the eminence of Deity and can be naturally participated in, but Deity cannot be participated in naturally. (Cf. below, pp. 138 ff.: The dignity of sanctifying grace.)