Wellington and Napoleon never met one another on the field of battle until they fought at Waterloo. Wellington's career during the Napoleonic Wars was focused mainly in the Peninsular where he fought several of Napoleon's best generals (Marshal Ney being one of them).

If one looks at a lot of the cartoons of the time most of them portray Napoleon pitted against either John Bull (the personification of Britain) or George III, as seen in this image.

So, in taking these two factors into account, we could say that they weren't viewed as rivals by the public until Waterloo had been fought and won by Wellington. Despite this, however, both leaders knew of one another's fighting capabilities and it is known that Wellington lauded over Napoleon, saying he presence alone was 'worth 40,000 men', whilst Napoleon viewed Wellington as a bad general and the British as ill trained soldiers. This gives a good and concise look into this.

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Interestingly, they never met in person and they also only fought on opposite sides of a battle once, at Waterloo. In the Peninsular War Wellington fought French marshals including Soult and Massena rather than Napoleon directly, while Napoleon's foes were mostly Central/Eastern European generals from Prussia, Russia and Austria (In fact it'd be interesting to know which general faced Napoleon in battle the most, if anyone knows?)

Here's a quote from each of them about the other.

Wellington spoke of Napoleon as an excellent general;

It is very true that I have said that I considered Napoleon's presence in the field equal to forty thousand men in the balance. This is a very loose way of talking; but the idea is a very different one from that of his presence at a battle being equal to a reinforcement of forty thousand men.

While Napoleon speaking to his officers before Waterloo was more dismissive;

I tell you Wellington is a bad general, the English are bad soldiers; we will settle this matter by lunch time.

However both of these quotes should be viewed in their historical contexts. Wellington was speaking after defeating Napoleon at Waterloo and would have had every reason to boost Napoleon's standing, to make his victory more impressive. Conversely, Napoleon was leading an outnumbered French army and attempting to boost their confidence immediately before the battle.

Certainly.

A great primary source for local British opinions of the era are the papers of various bluestocking women with strong connections to the court. Hester Piozzi described Bonaparte as "the Devil incarnate, the Appollyon mentioned in Scripture" while she showered Wellington with praise. The diaries of Elizabeth Montagu and Frances Burney are likewise similar.

The French were somewhat more conflicted in their treatment of Napoleon. Burney was married to a French émigré, the Count d'Arblay, and many French expatriates had mostly negative views of the First French Empire. The French media were even more fickle. During Bonaparte's 100 Days, Parisian broadsheets initially printed articles detailing Napoleon's return from exile in Elba as in the most excoriating of terms. By the time he arrived in the capital, articles of the era described his return in language reserved for the Rapture.

Wellington and Napoleon were certainly adversaries although the bitterness of their rivalry ebbed and flowed depending on the year and political conditions. At Waterloo, the only battle the two generals fought against one another, Napoleon chastised Wellington for his poor leadership on the field. Wellington was equally vitriolic, although both men had a certain level of restrained praise for their opponent. Wellington once said that "his presence on the field made the difference of forty thousand men" although this statement was made on retrospect after Napoleon's defeat at Waterloo.