[Poetry is based on a personal response. This is how I interpret the poem.]
It is important to note that Byron was a second-generation poet, one of several believed to embody the Romantic Movement in England, especially between 1800-1836. Of the many topics the Romantic poets wrote about, love was an important one, especially to Byron.
In essence, while Byron's "When We Two Parted" may seem to be simply about a break up, I believe that it ends in tragedy for the speaker. Several specific word choices move me to believe that she dies. (See "grieve" and "knell.")
There is not enough room here to summarize line by line, but I can summarize sentence by sentence in each stanza. In stanza one, the first sentence notes that the two (we can infer they were lovers) separated in heartbreak.
In silence and tears...
...infers that there was no closure to the relationship. Perhaps there was no discussion or even a note: it seems she cut him off, perhaps refusing to see him when he came to call or refusing to answer his letters. However, the hint for me of death begins with the idea that they have been apart for years, but that she began to grow ill—for...
Pale grew thy cheek and cold,
Colder thy kiss...
While she begins to look sick, "colder thy kiss" seems to imply the chill of death on one's skin after passing. There seems to be foreshadowing of this event, as he notes:
Truly that hour foretold
Sorrow to this.
Once she became ill, the sorrow he felt over the separation seemed to increase with a warning of impending death. In the second stanza, the dew "chill on my brow" may refer to waking on the day of the funeral, and that he feels it was a hint of how he now feels at the service—that the dew should have warned him of how it would impact him.
The vows that are broken may refer to pledges of love between them that were dashed aside when they separated. The light may refer to how she is revered in death—praised; but as people speak of her, he knows of their secret affair and that she was not as sinless as others might believe, so he feels shame in their shared, secret sin.
The third stanza refers to her name as a "knell" in his ear: a knell is a bell ringing, reflecting sorrow, usually death. Still after all this time, he reacts physically, with a shudder, wondering what it was about her that captivated him so. His relationship to her is secret; he notes he knew her beyond the casual:
...Who knew thee too well...
He states he shall rue her: "rue" means to "feel sorrow over...regret bitterly" or to "wish that (something) had never been done, taken place." However, I sense that his regret is not for the relationship, but for the love they lost. "Long, long" seems to indicate that he will mourn their parting and her passing for the remainder of his life: he suffers so deeply, he cannot express himself in words.
The last stanza offers a resolution to the poem: the speaker concludes that as they met in silence, so now he must grieve in silence as well. "Grieve" is a word often associated with death. He will never understand her...
...That thy heart could forget,
Thy spirit deceive.
When he speaks of meeting "after long years," I imagine he is speaking of when he dies and meets her again. Even over the length of his life, he seems to say that when he "greets" her, even in heaven, he will still mourn his loss "[w]ith silence and tears," because of his deep and unrequited love.