How did Augustine of Hippo feel about Jerome's Latin translation of the Bible (the Vulgate)?

Short Answer:
In the least, Augustine did not approve of Jerome's preferred translation of the Septuagint (he actually penned two, at least in part), specifically. At most he may have thought it a sin to even use it as scripture. His greatest fear is that it would lead to apparent discrepancies in opinion between Latin and Greek churches and grievously harm church unity. He did not object to a translation in general (in fact he very much wanted one), but to the source that Jerome choose, the Hebrew versions, instead of the Greek versions.

In a letter from Augustine to Jerome:

I beseech you not to devote your labour to the work of translating into Latin the sacred canonical books, unless you follow the method in which you have translated Job [which was by the Greek texts].

Some of the other translations Jerome provided where acceptable to Augustine. He even said this to Jerome in that same letter, which largely criticized the translation of the Septuagint:

At the same time, we are in no small measure thankful to God for the work in which you have translated the Gospels from the original Greek, because in almost every passage we have found nothing to object to, when we compared it with the Greek Scriptures.

Despite Augustine's passionate objection, Jerome's translation of the Septuagint from the Hebrew texts is today heralded as his best work and among many protestants (and somewhat Catholics) one of the best translations of the Old Testament.

Long Answer:
The issue that Augustine had with Jerome's translation was with his source material and was most definitely not about Jerome's ability. In fact, Augustine regularly praised Jerome's work.

But concerning news that Jerome was using the Hebrew versions of the text instead of the Greek for the translation, Augustine once stated in a letter to Jerome in AD 394:

I beseech you not to devote your labour to the work of translating into Latin the sacred canonical books, unless you follow the method in which you have translated Job, viz. with the addition of notes, to let it be seen plainly what differences there are between this version of yours and that of the Septuagint, whose authority is worthy of highest esteem. For my own part, I cannot sufficiently express my wonder that anything should at this date be found in the Hebrew manuscripts which escaped so many translators perfectly acquainted with the language.

Augustine thought it a waste of Jerome's efforts and ability to translate the Septuagint using the Hebrew texts available, believing that the Greek version was Divinely inspired. Using strong language, Augustine urges Jerome to use the same method for translating Job, which he approved of, which was to use the Greek and not the Hebrew text because the Greek was Divinely inspired in his opinion.

Jerome did not heed Augustine's advice, believing that the Hebrew was actually more accurate to the original text. He voiced his objection again to Jerome using the Hebrew in another letter in AD 403, although, he did praise the exhaustiveness of the work, but challenged the motive for the asterisks and obelisks in the translation. See the quote below:

I have since heard that you have translated Job out of the original Hebrew, although in your own translation of the same prophet from the Greek tongue we had already a version of that book. In that earlier version you marked with asterisks the words found in the Hebrew but wanting in the Greek, and with obelisks the words found in the Greek but wanting in the Hebrew; and this was done with such astonishing exactness, that in some places we have every word distinguished by a separate asterisk, as a sign that these words are in the Hebrew, but not in the Greek. Now, however, in this more recent version from the Hebrew, there is not the same scrupulous fidelity as to the words; and it perplexes any thoughtful reader to understand either what was the reason for marking the asterisks in the former version with so much care that they indicate the absence from the Greek version of even the smallest grammatical particles which have not been rendered from the Hebrew, or what is the reason for so much less care having been taken in this recent version from the Hebrew to secure that these same particles be found in their own places.

Augustine goes on in the letter, reveling his biggest concern with the source material chosen was a matter of Church unity. He feared that the Latin churches would have scripture that was too different than the Greek churches, which might lead to arguments and schism. He favored the Greek texts, on a logical basis, because it was already wide spread.

For my part, I would much rather that you would furnish us with a translation of the Greek version ... For if your translation begins to be more generally read in many churches, it will be a grievous thing that, in the reading of Scripture, differences must arise between the Latin Churches and the Greek Churches, especially seeing that the discrepancy is easily condemned in a Latin version by the production of the original in Greek, which is a language very widely known; whereas, if any one has been disturbed by the occurrence of something to which he was not accustomed in the translation taken from the Hebrew, and alleges that the new translation is wrong, it will be found difficult, if not impossible, to get at the Hebrew documents by which the version to which exception is taken may be defended.

Augustine even discusses a certain bishop who 'narrowly escaped' a 'calamity' where he might have lost his congregation forcing the bishop to 'correct [Jerome's] version in that passage as if it had been falsely translated.' Augustine's concern for Church unity and reduced confusion is summed up thusly:

You would therefore confer upon us a much greater boon if you gave an exact Latin translation of the Greek Septuagint version: for the variations found in the different codices of the Latin text are intolerably numerous; and it is so justly open to suspicion as possibly different from what is to be found in the Greek, that one has no confidence in either quoting it or proving anything by its help.

Jerome, however, did not respond favorably and was even somewhat hostile to Augustine's reasoning, which required that Church unity be more important than the exactness of the translation to the original writings. His letter to Augustine in AD 404 states:

you ask why a former translation which I made of some of the canonical books was carefully marked with asterisks and obelisks, whereas I afterwards published a translation without these. You must pardon my saying that you seem to me not to understand the matter: for the former translation is from the Septuagint; and wherever obelisks are placed, they are designed to indicate that the Seventy have said more than is found in the Hebrew. But the asterisks indicate what has been added by Origen from the version of Theodotion. In that version I was translating from the Greek: but in the later version, translating from the Hebrew itself, I have expressed what I understood it to mean, being careful to preserve rather the exact sense than the order of the words. I am surprised that you do not read the books of the Seventy translators in the genuine form in which they were originally given to the world, but as they have been corrected, or rather corrupted, by Origen, with his obelisks and asterisks; and that you refuse to follow the translation, however feeble, which has been given by a Christian man, especially seeing that Origen borrowed the things which he has added from the edition of a man who, after the passion of Christ, was a Jew and a blasphemer. Do you wish to be a true admirer and partisan of the Seventy translators? Then do not read what you find under the asterisks; rather erase them from the volumes, that you may approve yourself indeed a follower of the ancients. If, however, you do this, you will be compelled to find fault with all the libraries of the Churches; for you will scarcely find more than one manuscript here and there which has not these interpolations.

In AD 405 Augustine replied with the same argument and begged Jerome to deliver the translation to him so that it may not be used in any authoritative church gathering. As far as historians know, Jerome did not respond and continued with the translation.

SOURCES (In order of use)
A selection of the letters between Jerome and Augustine
Septuagint Wikipedia Article - Section on Christian use of
Jerome Wikipedia Article - Section on Translations and Commentaries
Jerome Wikipedia Article - General Reference