Paris in the Merovingian apogee (Vth to VIIth centuries) was a rather noticeable city, even if having decreased in size since the Late Antiquity : after the chaotic period of the mid-to-late Vth century, the early Merovingian kings made it, not as much a capital than their main city.
Set on a rather favourable emplacement on the navigable Seine river, the city also preserved some apparatus of being a secondary imperial residence in Gaul, while not overly distinct from other comparable agglomerations North of the Loire. Still, the prestige Paris gained from its local elites (as Ste. Genovefa) who had good relations with the new dynasty since one or two generations, its situation both in Gaul and in the rough middle of Clovis' kingdom made the king electing to set his residence in the former imperial palace, itself set on the former baths.His successors would keep giving Paris and its countryside a special attention, holding these in indivision rather than splitting it among themselves, connected to their own relatively nearby courts, and as Clovis did, supporting the foundation of many religious establishment (for example, Ste. Geneviève, St. Vincent and of course St. Denis as a royal necropolis), along with other activities tied to royal patronage and authority, such as jewel-making, monetary workshop, trade (notably connecting the Channel trough the Mediterranean basin, the fertile plains of Northern Gaul and Gallic productions as a whole)
There's a drastic lack of sources on the city life itself, known from punctual royal and episcopal interventions, but it wouldn't be fantasist see Paris as a relatively dynamic agglomeration, gathering maybe 15 to 20 000 inhabitants (in what became l'Île de la Cité but also on the banks, mainly the left), and benefiting from a special royal attention. By the late VIIth century, however, the city declined in importance in favour of Rhenan or Meusan royal residences, and while still considered the head city of Francia, would recover only its position with the early Capetian period.
Late Latin would have primarily heard into the city, as in the rest of Frankish Gaul, existing in two distinct but related and constantly interacting registers.
A sermo politus, that is a proper Latin, would have been expected of Paris' elites, that is bishops, clerks, royal agents, aristocrats, etc. At this point, it obviously wasn't the Classical Latin of Cicero or even Augustine, but a formal Latin (sometimes called 'Merovingian Latin') that while keeping ancient references, would have evolved along popular speech in pronunciation or even syntax and lexicon.
The bulk of the population, on the other end, would have spoken a colloquial register of Latin : the sermo rusticus, people's Latin, was more directly issued from the Popular Latin of the late Empire, a probable ensemble of accents, barbarisms, regionalisms, etc. that would be mostly be "unchecked" by the lack of written fixations.
As distinct they were, both of these variations were essentially the same language, evolving as Early Romance and, in this case, Old French, as it emerged ca. 750. As it happened, however, there was a growing differentiation, with mutual understandability being exacerbated by the Carolingian reforms of chancery Latin and the stress put on returning to 'proper' pronunciation and syntax, with the Reichenau Gloses highlighting these discrepancies, eventually leading to the absence of mutual understandability between the 'renovated' Latin (which became Medieval Latin) and colloquial Romance (which became Old French) stated in 813 in the Council of Tours.
Frankish language(s) probably remained in existence in the Parisian basin during the period, but unwritten and likely largely as a sociolect (that is a language used and shared among members of a same social group rather than ethnic) : the rather limited numbers of Vth century Franks and their immediate descendents (augmented by the 'Barbarisation' of northern Gallic elites in the VIth century) certainly prevented it to be widely spoken then.Still, an at least formal and legal knowledge of it could have been heard, in the right social circles.
Beyond that, Paris being relatively prosperous but probably not having the bearing of a cosmopolitan city, the occasional hearing of Saxon (from communities settled in Normandy and Picardy), Greek (from 'Syrian' traders) or forms of Old High German from the troops and nobles from the peripheral peoples under Frankish overlordship in Germania (Thuringians, Alamans, Bavarians, etc.) wouldn't be impossible, but rare occurrences.
If you were a royal agent of any sort (comes, missus, bishop, etc.) or a noble travelling around in Gaul in the same period, there wouldn't have been much difference apart from Frankish virtually disappearing south of the Loire and being more present east of the Meuse and north of the Somme.
On the other hand, Latin in either of its forms outside the former Empire would be at best a political language used exclusively by your peers.
In Gaul itself, at most could you be acquainted with Old Briton in westernmost Armorica or Old Basque in Aquitaine if you were sent to fight raiders or manage nearby lands, or with the last speakers of Late Gaulish in some regions' countryside (why you'd get in contact with the these would be anyone's guess, however) especially in the highlands (Massif Central and Alps).
A last language, while socially and culturally marginalized, could be heard in the realm : Hebrew as a religious and prestige language in Gallic Jewish communities (especially present in southern Gaul but also in the North, including Paris or Orleans). Their situation could be precarious, having a legal existence, kings, bishops and urban population finding craftsmen, traders or even agents among them (although for a brief period), but in the same time under popular, episcopal and royal duress.