I'm not sure who'll have the biggest chance to advance as I havn't been paying much attention to international competitions. My prior-to-the-draw expectations. But given my lack of viewing their matches, I might be way off though.
Ajax
Honestly, they don't give a crap who they're playing right now, so I'd say they stand a decent chance against any team that can motivate them to give it their all. The dirtier the opponent plays in terms of fouls and the more defensive they play the lower the chances I'd give Ajax.
Mostly because Dutch league players get frustrated and provoked relatively easily when faced with getting kicked and dives. Hence I'd be more concerned about facing say Porto or Juventus, than I'd be with a team that likes to play ball. Ajax' 'youth' teams shown in the past they could dance with ManC and Barca and this team is a bit more mature mentally than the one facing Man Utd in the finals of the Europa League.
The KNVB is openly helping Ajax with the league schedule to get the players sufficient rest and allow to prepare for the matches, so there's that too I suppose.
Barcelona
I would say Barcelona stands a decent chance given its comfortable lead in the domestic competition and being able to rest their players and concentrate on the CL - except that they play Atletico on the 6th of april. Then again, leading so much might make them more complacent, especially if they beat Atletico. For them, the second leg is probably easier to prepare for. Either way, I'd see them reach the finals before anyone else.
Juventus
Fluked a bit and really unpredictable tbh, their results haven't been consistent at all and they've lost quite a few matches this CL already, including to Young Boys. I'd not rate them as the top contenders.
Liverpool
Can't say I've been impressed by them. Exceptionally accurate long ball by Virgil van Dijk, but... that's just a long ball forward. Not really impressed with Liverpool's capacity to create and use chances this CL. Frankly I doubt they'll go another round regardless of opponent.
ManC
I'd say one of the big contenders this season. Scores a lot and hard to beat, but can be beaten on a bad day. Would imagine they're going to the next round, unless they meet Barca, then all bets are off.
Man Utd
Speaking of fluking. Also quite inconsistent results, lost quite a few matches in the CL so far. Not a big amount of goals made or against in most these matches being won or lost on a one goal difference. I'd give them a 50-50 chance to move to any next round, regardless of opponent.
Porto
I'd never trust a Portuguese to play completely fair and thus get under their opponent's skin (bit of prejudice here in that regard given experiences with other Portuguese teams), which is actually a great quality for cup fighting tbh if you can create mental unrest with your opponents. I'm just not sure about their quality as I havn't seen them play, but they seem to score 1 to 3 goals every match. Possibly been a bit lucky with the strength of opposition so far though, so curious how that'll go.
Tottenham Hotspur
Eh. Ajax defense. :P Pretty stable positive results overall against varying opponents. Slightly above average chance of moving to the next round I'd say.