First, I'll start my answer by splitting it up into Arena vs Constructed.
Arena drafts usually favor players who manage to draft a reliable and strong curve, often forcing tempo onto the opponent. Mana costs for removal spells vs the minion cost itself also matters, which is the reason why Mage is considered one of the best classes in Arena due to the high number of efficient clears (Frostbolt, Fireball, etc.), and so is Rogue (Backstab, Eviscerate, SI:7 Agent etc.).
Another important factor that also comes into play is the class's hero power. The classes which see the most success in Arena (Mage, Rogue, etc.) all have extremely efficient hero powers that allows them to efficiently control the board and build it to their favor. Paladin hero power is also useful in a sense that although it is not a "ping" type hero power, the flood of 1/1s will definitely help in establishing an early board control therefore gaining tempo.
Now, looking at the Priest class hero power, we can see that it is extremely useful in maintaining board control (e.g. your 2/3 minion will be able to trade into enemy's 2/3, and you can then use your hero power to heal the damaged minion back to full health, therefore giving u effectively free damage to enemy minion). However, this only applies when you actually do have a minion on the board. Without a minion on the board, it is a completely useless hero power, even worse than the Warrior's Armor Up or Hunter's Steady Shot. This may often result in a situation where you are forced to do nothing on turn 2 or even turn 3, therefore losing you huge amounts of tempo. This will eventually result in a loss simply due to the tempo loss, and as I have mentioned, Arena is all about gaining and maintaining tempo.
Another factor that comes into play in Arena is its class-specific cards. Just a few months ago, Paladin was on the top of the pack in Arena, almost on par with Mage. This was simply due to the extremely powerful class-specific cards it had at its disposal, such as the Shielded Minibot (which allowed it to trade potentially with 2 2-mana minions while costing only 2 mana themselves), Truesilver Champion (effectively a 4 mana weapon that can clear up to 2 separate minions that might cost up to 3-4 mana), and other various cards such as Muster for Battle and Aldor Peacekeeper. In addition, the class-specific legendary for Paladin, Tirion Fordring, could be considered the most powerful legendary in Arena.
Whereas, when we look at Priest-class specific cards, we often see cards that are far too specific and situational, and a lot of them are tech choices that might not be good in all situations. Just think of Silence and Purify, which can be really powerful card when used in an appropriate situation, however it has equal chances of being a dead card in your hand.
All in all, with all these disadvantages, Priest is indeed proving to be a bad choice of class for Arena.
Priest is indeed extremely bad in constructed play at the moment. This is not only seen in just normal ladder, but is also apparent from the choice of decks in professional tournament scenes.
Although it was once one of the best Control type class (Dragon Priest and Control Priest had easily one of the highest win rate in ladder some time ago), it is now considered far too inconsistent to be played on the ladder. This was mainly due to the high value cards it lost with the introduction of Standard format, as well as the introduction of the Whispers of the Old Gods cards, which gave Shamans and Warrior extremely powerful cards whereas in comparison, Priest received extremely little.
Firstly, the Standard format. With the Standard format, Priest lost many of its valuable cards such as Shrinkmeister (a shrinkmeister-Cabal Shadow Priest combo could potentially steal even a Ysera). Not only that, but various board clears such as Lightbomb was removed from Standard, which hurt Priest by quite a large amount.
Secondly, the Whispers of the Old Gods patch. This patch introduced many different cards, many of which has found a place in the meta. For example, Shaman got a 4 mana 7/7 Overload (2), which is extremely potent tempo play, and the overload effect synergies with cards such as Tunnel Trogg well as well. Whereas in contrast, Priest got almost nothing out of this patch. This has affected Priest's win rate in general, especially in a meta that prefers aggro-archetype decks and tempo-orientated decks in general.
Another issue that comes with Priest is how heavily Priest relies upon proper card draws to win games. Card draws itself is a RNG-based mechanic, therefore making it unreliable. This is especially evident in the recent Resurrect Priest deck, which is highly unreliable due to its RNG. In addition, if you manage to get a strong turn 3 or 4 minion to play, maybe something of like 3/6 or 2/5 or whatever, you can continuously trade and control the tempo while repeatedly healing the minion for the damage taken. However, outside of that, tempo is heavily on the opponent's favor, therefore resulting in the low win rate.
Last but not least, I would like to mention another issue. Priest has extremely good anti-aggro and anti-control cards, however, the problem here is that in order for your deck to be anti-aggro or anti-control, you need to build your deck in that manner. You have to choose one road, and this results in a situation especially on ladder where you don't know what the opponent is playing and your deck might not be suited to be played against it. This further lowers Priest's consistency on the ladder, thereby making it a weaker class in comparison.
Don't lose all hope yet, though. With the introduction of some new cards with Karazhan, Onyx Bishop as well as Priest of the Feast are looking relatively good, and might actually bring Priest back into the meta, especially in the form of Resurrect Priest and Silence Priest.
Based on the current meta as of 15/09/2016, Karazhan patch