The Manchu Qing was absolutely cognisant of its Jurchen Jin predecessor, and did, to a certain extent, seek to portray itself as its successor. That said, it did so primarily in the period from 1616 to 1636, when the Qing was still the (Latter?) Jin, and the Manchus still the Jurchens.
While the exact rationale behind the naming of both the Latter Jin and Qing states has never actually been firmly established, it is almost certainly not coincidence that Nurgaci chose to refer to his state in 1616 as the 'golden state' (aisin gurun), just as Agūda had named his state the 'golden state' (anchun gurun), both of which would be rendered as Jin 金 in written Chinese. (Whether the term 'Latter Jin' (hou Jin in Mandarin) was used in any official capacity during the lifetime of the state itself, which would have made a claim to continuity explicit, is unfortunately uncertain.) But such emphases on continuity were not a consistent feature of Manchu historiography. In the first surviving version of the Manchu foundation myth from 1635, the Aisin Gioro clan's progenitor, Bukūri Yongšon, who was almost certainly fictional, was said to have been immaculately conceived by the spirit maiden Fekulen, who commanded him to take the leadership of the Jurchens, implicitly in the time of Ming rule in China. Only in later versions of the myth was it made explicit that these Jurchens had previously been those under the Jin empire, but in early ones, that continuity was not stated in the text.
As Pamela Crossley notes, there are some interesting parallels between the fabricated story of Bukūri Yongšon and the actual history of the Gioro clan(s), which in turn highlight some of the complexities . In the myth, Bukūri Yongšon is an outsider to the Jurchens whose claim to power derives solely from his divine birth. And the official history of the Jin noted that the Gioro had originally been members of the Hurka tribe, and were thus considered non-Jurchens and ineligible to marry members of the ruling Wanggiya clan, only later being formally recognised as part of the Jurchen people. It was thus absolutely in the interest of Nurgaci and Hong Taiji to demonstrate some concrete point at which their clan was formally integrated into Jurchen society, which in the absence of a strong written record would have to be fabricated and thus led to the Bukūri Yongšon myth. But it was also in their interest, and especially that of Hong Taiji, to create some kind of identity that cohered around the Aisin Gioro in particular, which likely is what ultimately motivated Hong Taiji to rename his people the Manchus, and to subsequently retcon Jurchen history by claiming 'Jurchen' was a pejorative label to obscure the Manchus' true name. Bukūri Yongšon serves as a somewhat paradoxical pivot point for the Qing narrative of Jurchen history: on the one hand, he ties the Aisin Gioro into a history of the 'Manchu' people that was explicitly said to stretch even further into the past; on the other, he serves as the genesis of Manchu history by being the ancestor to the imperial clan around whom the construction of Manchu identity would theoretically revolve going forward.
Bukūri Yongšon ends up as a useful illustration of the somewhat complicated relationship the Qing, throughout their existence, held to the Jin. The Jin could be looked to as an earlier phase when the Manchus' ancestors held great power, and therefore the Qing could reasonably seek to tie itself into that history. Yet the rulers of the Qing were not directly descended from those of the Jin, and had an interest in creating a basis of rulership that centred specifically on themselves. Privately, the Qing did not see the Jin as having been some sort of perfect entity; indeed, it looked on the Jin at least partially as a failure. But it did so in a way that paralleled their assessment of the Mongol Yuan. When the Manchus began translating the Chinese dynastic histories into Manchu under Hong Taiji, the first three were to be of fellow 'conquest dynasties' – the Khitan Liao, the Jurchen Jin, and the Mongol Yuan. The Jurchen Jin, especially, was of interest to Hong Taiji, who explicitly recognised its antecedence to the state now under his rule. While not necessarily stated explicitly by Qing rulers, they may well have drawn the lessons of these states as follows: a state could not afford to break from the neo-Confucian patterns of rule to which the people of China were accustomed without fatally compromising their rule in China (the lesson of the Yuan), but neither could it allow the integrity of its core ruling group to falter, which would be equally compromising (the lesson of the Liao and the Jin). The Jurchen Jin, in the view of the Manchu Qing, had fallen to the Mongols because it had failed to maintain the integrity of the Jurchens and prevent acculturation to Han Chinese ways. It was to be emulated in some regards, but repudiated in others.
But despite understanding certain failings of the original Jin, the Latter Jin didn't see it as being worthy of completely discarding, indeed far from it. In 1622, Nurgaci was supposed to have made a speech to his troops where he claimed that the great cities of China were not immutably Chinese, because the Jurchens had held many of them in the past. The Jin were not specifically named, but would of course have been the state under whom that had taken place. In 1629, during the only raid that Hong Taiji would ever lead into China proper, he stopped at Fangshan to make sacrifices at the tombs of the Jin founder Agūda and the fifth Jin emperor Shizong. Latterly, he would revise the Manchu histories to make explicit parallels between several aspects of Agūda and Nurgaci's lives and careers: both having their lives saved by Han officials at one stage, both initially serving the interests of a Han-led empire before being forced to rebel, and both uniting and reorganising the Jurchens into a people capable of turning that rebellion into conquest. Despite their ultimate fall, the Jin still served as an example for the Manchus as a period of past glories.
However, the emphasis on Jin heritage under Nurgaci and Hong Taijin came to be softened as circumstances changed. Needing Ming non-interference in his wars with the Chakhar Mongols, in 1631 Hong Taiji, in a letter to the Ming emperor, declared that he was not aiming to replicate the circumstances of the Song, and that just as the Ming were not the Song, his state was not the old Jurchen Jin. In turn, he would proscribe the term 'Jurchen' in favour of 'Manchu' in 1635 and discard 'Jin' for 'Qing' the year after. Descent from the old Jurchens was, as stated, far from outright repudiated, but after 1636, the newly named Manchu Qing state was one that ceased to imply the restoration of the 12th-13th century Jin, but rather a new state whose ties to the Jin were chiefly historical and genealogical. This was further compounded when Lighdan Khan, the Chakhar khan who was the last serious claimant to the title of Yuan emperor, surrendered to Hong Taiji after being hemmed in by revolts from within and by attacks from the Jurchens. The two aforementioned renamings occurred in the wake of the absorption of the Chakhar khanate over the course of 1634-5, and Crossley rather pointedly suggests that the Qing ought to be seen less as a direct continuation of the Latter Jin, and more a union of Latter Jin with the Northern Yuan, inheriting not just the 12th century Jin legacy, but also that of the Mongol Empire. From here on out, the Jin would move to a position of lesser importance as Qing rulers worked to build a more uniquely Qing-specific ruling ideology, with distinct variations for Manchus, Mongols, Han, and latterly Tibetans and Muslims.
We have focussed so far on the Qing self-image as presented to the Manchus, but for the Han Chinese, however, the Jin were less of an ideal object to harken back to. For instance, the Yongzheng Emperor, in his dayi juemi lu, a propaganda text attempting to defend the Manchus against charges of ethnic favouritism through claiming that the Manchus had been successfully acculturated to Han ways, only mentioned the Jin in passing when discussing major periods of disunity between northern and southern China. In this, he merely discussed the existence of mutual, ethnically charged animosity between northerners and southerners in these periods as being of equivalent nature, and did not attempt to provide a specific defence of the Jin as a state. As noted, the Qing had, even to the Manchus, stopped portraying itself as a direct resumption of the Jin as opposed to its own state, so within China there was little motive to directly associate the Qing with a period widely perceived as one of unjust barbarian dominion.