This is a slightly complicated subject because our modern world view can cloud how we perceive historical ideas. I apologise in advance, as I'm going to include lots of background on the ancient world view which tended to lean east, rather than westwards, as your question asks.

Nonetheless, as /u/lilmonstertruck has mentioned, they certainly wouldn't have viewed the Atlantic ocean in the same way that we do today. The whole concept of 'world' view is a particularly interesting subject in itself especially as it develops through different cultures, civilizations and eras.

But back to the case in hand. With any moment in history, the best way to understand it is usually to investigate precedents leading to that moment. In this case, I think the best way is to refer back further than Caesar and look at how the Greeks viewed the world as a whole. The Romans based and derived a great deal of their culture and intellectual culture from the Greeks (perhaps typified by the Library of Alexandria).

I think the best and earliest place to start would be with Herodotus. His work The Histories encompassed a great swath of the known world, with Greece in the middle, and shows one way of seeing the ancient perception of the world.

To sum-up what is explained in the Livius.org article:

[Herodotus] believes that the world is symmetrical. When he compares the Danube and the Nile, he points out that they both divide a continent in two halves, and that their deltas are on the same geographical longitude. Since it is -according to Herodotus- well-known that the sources of the Danube are in the far west, in the Pyrenees, he is positive that the sources of the Nile must be in the west too, in the Atlas mountains. And because the Danube has five mouths, two of the seven branches of the Nile must be canals. This symmetry is not complete: Herodotus knows that it is cold in the north and warm in the south.

Between these extremes is Greece at the midpoint of the earth. The fact that there is an unpleasant climate at the ends of the (flat) earth, is compensated by the natural richness of those far-off lands: cinnamon, gold, tin, amber and all the perfumes of Arabia (text).

This provides an earliest impression of how the world was seen. What is interesting (although not that surprising) is how much more knowledge there is about Eastern areas going as far as India. This is unsurprising as the landmass was continuous and sea-trade routes much more accessible. Thus for the Greeks, the 'Atlantic' was far flung and on the edge of the world (note the tin islands in the far North-west which we would now recognise as the British Isles). The world view changed again under the conquests of Alexander and his ranging expeditions far into eastern regions. The extent to which Alexander regarded the world is exemplified in the reported speech in Arrian (a Greek/Roman historian)

'For a man who is a man, work, in my belief, if it is directed to noble ends, has no object beyond itself; nonetheless, if any of you wish to know what limit may be set to this particular campaign, let me tell you that the area of country still ahead of us, from here to the Ganges and the Eastern ocean, is comparatively small. You will undoubtedly find that this ocean is connected with the Hyrcanian Sea, for the great Stream of Ocean encircles the earth. Moreover I shall prove to you, my friends, that the Indian and Persian Gulfs and the Hyrcanian Sea are all three connected and continuous. Our ships will sail round from the Persian Gulf to Libya as far as the Pillars of Hercules, whence all Libya to the eastward will soon be ours, and all Asia too, and to this empire there will be no boundaries but what God Himself has made for the whole world.

Alexander (excusing a bit of dramatic hyperbole) certainly seems to think that the conquest of the whole world is within his grasp. Had his army not eventually rebelled, he may well have continued east to discover how wrong he was.

The reference to Alexander here is particularly pertinent as Caesar was often held in comparison to Alexander as a matter of course. Plutarch compares the two in his work Parallel Lives

But unlike Alexander, Caesar extended his conquests west and north. This did bring him into contact with the Atlantic (in a very limited manner). From documents such as Caesar's Gallic War (the whole of which can be read here) The Roman relationship with the Atlantic was that they did not know what was beyond it, and where not really that interested. The Atlantic presented no obvious opportunity for economic or military prestige and Britain was already considered the edge of the known and habited world. There is some discussion about how much the Romans knew about, and considered conquest of Ireland. But while there was some knowledge, the Romans appear to have seen no real benefit in extending the empire into another rainy and cold province.

So to sum up, (and this is by no means a definitive answer) the Romans did not know what was on the other side of the Atlantic and they did not have much of a desire to find out. The Atlantic was simply too large and the Eastern regions too politically important and resource rich for the Romans to give much credence to and serious expeditions west.

Some more academic-y reading that you might find interesting:

McLaughlin, R. (2012). What did the Romans (N)ever Do For Us? History Ireland, 20(2), 16–19. doi:10.2307/41480783

Lynch, J. J. (1992). The Maps of Discovery. The North American Review, 277(5), 6–15.