We are taught that the Holy Ghost is a cleanser (e.g., Moroni 6:4) - a purifier. It appears that we cannot purify ourselves - that we must be purified by regular exposure to the Holy Ghost. Also, it is when our hearts are connected to the true vine that they are able to bring forth fruit meet for repentance (John 14:1-5; Alma 13:13) - which leads to baptism, which leads to receiving the gift of the Holy Ghost, which leads to the companionship of the Holy Ghost, which leads to an infusion of the Spirit of God, which leads to an enlivening connection to God, which leads to "doing the will of God", which leads to becoming like God, which leads to seeing God - not only eventually in the next life, but fundamentally in each other as we become more like Him.
In the end, as I John 3:2 states:
"Beloved, now are we the sons [and daughters] of God, and it doth not yet appear what we shall be: but we know that, when he shall appear, we shall be like him; for we shall see him as he is."
Being connected to God through the Holy Ghost will allow us to do more than see Him "in all that is around us;" it literally will allow us to see Him "as He is".
In summary, becoming pure in heart means being connected to the cleansing power of the Holy Ghost - and the result of that purity is the ability to see God's will for us and literally, in one way or another and in this life or another, see God as He is. The key, as stated at the beginning of this post, is to pray diligently to be cleansed and hearken (hear and follow) unto the promptings of the Holy Ghost at all times - to repent by becoming more and more like Him as we go through life - to become as He is.
Thus, having a purer heart and cleaner hands is the very foundation of having his image in our countenances - as those things are what help us become like him.
The Jews rejected Jesus' statement that those who had seen Him had seen His Father as the height of blasphemy, and modern Christianity still rejects Mormonism's claim that "as God is, man may become" - but that is the principle taught in the Beatitudes. Truly, the greatest paradigm-shattering aspect of the Gospel of Jesus Christ is that we can become joint-heirs with Christ - that we may press forward with faith in the hope that we shall see Him as He is for we shall be like Him - that, someday, we will see God in the mirror.