I can't remember any instance of Gollum singing a song (certainly not one that doesn’t involve fish), but the lyrics seem to reflect Gollum’s history and a sense of being an outcast.
Where once was light
Now darkness falls
Where once was love
Love is no more
The first verse could be referring to his transition from living in an extended family by the river to his fall to the lure of the ring, expulsion and subsequent life alone in a cave under the Misty Mountains.
And we will weep
To be so alone
We are lost
We can never go home
The ‘we are lost’ motif may be picking up on Gollum’s response to Faramir:
‘We are lost, lost,’ said Gollum. ‘No name, no business, no Precious,
nothing. Only empty. Only hungry; yes, we are hungry. A few fishes,
nasty bony little fishes, for a poor creature, and they say death. So
wise they are; so just, so very just.’
Chapter 6. The Forbidden Pool. The Lord of the Rings (The Two Towers, Book 4)
There’s a sense of loneliness and betrayal in the song, which reflects Gollum’s perception. He does show signs of trusting Frodo:
‘Nice Master!’ he whispered. ‘Nice hobbit, come back to poor Sméagol.
Good Sméagol comes. Now let’s go, go quickly, yes. Through the trees,
while the Faces are dark. Yes, come, let’s go!’
And feels betrayed when he’s caught by Faramir’s men.
Gollum crawled along close to the brink for a little way, snuffling
and suspicious. Presently he stopped and raised his head. ‘Something’s
there!’ he said. ‘Not a hobbit.’ Suddenly he turned back. A green
light was flickering in his bulging eyes. ‘Masster, masster!’ he
hissed. ‘Wicked! Tricksy! False!’ He spat and stretched out his long
arms with white snapping fingers.
From Gollum’s point of view he’s been betrayed by Frodo, and Frodo recognises this.
‘Yes. Well no, I didn’t get him. He came to me, because he trusted me
at first, I’m afraid. I did not want him tied up like this. I hope it
will be all right; but I hate the whole business.’
If the song is from Gollum’s point of view, then this could fit with the verse:
These tears we cry
Are falling rain
For all the lies
You told us
The hurt, the blame
The lyrics could be directed at Gollum (because he’s lied) but the song seems to be more a lament for Gollum’s descent.
These tears you cry
Have come too late
Take back the lies
The hurt, the blame
This could be a reference to Gollum’s near repentance that’s disrupted by Sam on the stairs.
The gleam faded from his [Gollum’s] eyes, and they went dim and grey,
old and tired. A spasm of pain seemed to twist him, and he turned
away, peering back up towards the pass, shaking his head, as if
engaged in some interior debate. Then he came back, and slowly putting
out a trembling hand, very cautiously he touched Frodo’s knee — but
almost the touch was a caress. For a fleeting moment, could one of the
sleepers have seen him, they would have thought that they beheld an
old weary hobbit, shrunken by the years that had carried him far
beyond his time, beyond friends and kin, and the fields and streams of
youth, an old starved pitiable thing.
Chapter 8. The Stairs of Cirith Ungol. The Lord of the Rings (The Two Towers, Book 4)
Gollum is lost (spiritually), alone and friendless, and he chooses not to change.
So in the end
I will be what I will be
No loyal friend
Was ever there for me