One of the interesting things about Garrosh Hellscream's rise to the position of Warchief of the Horde has been his display of leadership. When he was in charge of the Warsong Offensive, he lead the Horde in battle against the forces of the Lich King's Scourge and the Alliance. Upon returning from Northrend covered in glory, he began his tenure as Warchief by unleashing Horde forces to seize control of Azshara, make inroads in Ashenvale and Stonetalon, and encouraged (one could even say strongarmed) the Forsaken under Sylvanas Windrunner into invading Gilneas in order to secure a port for future advances.
Following this, he orchestrated an elaborate plan to lure the Alliance's leadership to defend Theramore, a plan that nearly succeeded in destroying them with a mana bomb, and has since extended the war to Pandaria where he's dedicated a great deal of the Horde's military resources (and those of factions within the Horde like the Blood Elves) towards finding ancient weapons or things that could be made to serve as weapons, all while sending his Kor'kron to occupy the Echo Isles. In short, Garrosh has been a dynamic leader, and it's hard to dispute that he's made more gains for the Horde during his time as Warchief - both the Dragonmaw and the Blackrock have become powerful Horde allies under Garrosh, Gilneas is heavily contested, the Forsaken have spread across northern Lordaeron, Azshara is firmly under Horde control.
Yet, it appears (to this writer, at least) that Garrosh's main problem as a strategist is that he loves the big, grandiose scheme. Garrosh, rather than set himself on a path to consolidate gains, attempted to lure the leadership of the Alliance to Theramore so he could blow them all up at once, and instead failed to do more than give them a rallying cry and an atrocity to stiffen their resolve. The cost in troops and material was significant, the weapon used during the attack was taken from Horde hands and cannot be used again. This led Garrosh to attempt to find a substitute in the Divine Bell, which also ultimately rebounded in failure. While a skilled war leader, capable of motivating his troops and leading them to victory, Garrosh as a general displays no patience for the logistics and demands of war - his dismissal of Varok Saurfang's concerns during the planning of the Warsong Offensive maintains itself today.
In contrast, one of Garrosh's great strengths as a strategist is that nothing is unthinkable to him - he'll ally with tribes that were so far beyond the pale of the Horde that they'd elected their own Warchief, he'll create or investigate weapons of mass destruction on a scale usually seen by entities like Arthas or Deathwing, he'll countenance attacks on Alliance troops while they're engaged with the Scourge. He's opened not one but two additional fronts for the Horde outside of the Kalimdor battlefields he was already engaged in, and it's generally worked out for him. Garrosh knows the value of surprise, even if he does tend to rely on it too much.
While Garrosh struggles as a political leader due to his autocratic mien and unwillingness to listen to criticism or contrary ideas, this isn't a problem with Hellscream the general. Horde troops, especially orcs, respond well to a commander who leads directly and through clear orders he expects to be immediately followed. This is another strength of Garrosh as a military leader - he can not only motivate his troops, but he understands them. However, it's also a weakness when dealing with non-orc troops in the Horde war machine. One of Hellscream's biggest failures as Warchief is his absolute inability to use the forces at his command to their utmost. He can motivate and lead orcs, and to a lesser degree tauren (mainly due to the respect he shows their capacity as warriors) but he is much less skilled at not only commanding but understanding how to make use of the other military forces of the Horde. His use of the Forsaken during the original push into Gilneas was so aggressively unwilling to consider the costs of his decisions that it actually motivated Sylvanas to return from her own suicide to prevent Garrosh from wiping out her soldiers. Garrosh shows a similar, although less pronounced, disdain for the trolls, blood elves and goblins under his command.
In short, Garrosh's leadership style works very well with orcs, and not nearly so well with anyone who is not an orc. This is magnified by his thinly-veiled contempt for his non-orcish faction leaders (which isn't helped when, like Vol'jin, they give it right back to him) and his willingness to appoint people like Malkorok of the Blackrock to leadership of the Kor'kron, which has become much more of a personal bodyguard under Garrosh.
Garrosh is tactically brilliant. Given an objective, he can and will come up with a way to achieve said objective if it is at all possible to do so. He succeeded in breaking through the heavy Scourge presence in the Borean Tundra and established Horde bases throughout the Dragonblight, and after the events of the Wrathgate successfully led the Horde war machine into Icecrown Glacier and the ultimate showdown with the Lich King's forces. Since that time (when he was aided by old veterans like Varok Saurfang and outside forces such as the Ashen Verdict) he's proved himself capable of devising plans that allowed the Horde to push deep into former night elf territories, keep the Alliance bogged down in the Southern Barrens and it cannot be overstated how close he came to completely wiping out many of the Alliance and Kirin Tor's leaders in Theramore. The problem isn't with his tactics, but rather, his strategy. Namely, he doesn't have one.
There is no overarching horde war plan - they have no directed intent and no design for how to win the war with the Alliance. Garrosh wants to take control of Kalimdor but aside from schemes like Theramore and the Mana Bomb, he has no real strategy plotted out for how to do this, and no way to hold any of the gains he's made during the Cataclysm. In short, Garrosh Hellscream is an excellent tactician, and a talented leader of orcs, but he's a terrible general. He kills his subordinates for failing to understand what he means instead of what he says and as time passes, and the war between the Horde and Alliance becomes entrenched, Garrosh's over-reliance on flashy surprise gambits and taking advantage of his enemy being preoccupied with other problems (be it ambushing Alliance troops in Icecrown who were fighting the Scourge, or using the Cataclysm to push into Ashenvale) will become untenable, and he'll be forced to fight a war that he has no patience for, one that demands incremental victory and planning to sustain your gains.
He needs to learn how to respect and motivate more than just orcs, how to make plans to complete small objectives and not just grandiose schemes. He needs to lead the entire Horde, and lead it into war, and not just from in front of his troops but also from a political perspective. Garrosh has learned a great deal during his time as Warchief, but this war will either teach him to compromise, or he'll lose it.
While you don't need to have played the previous Warcraft games to enjoy World of Warcraft, a little history goes a long way toward making the game a lot more fun. Dig into even more of the lore and history behind the World of Warcraft in WoW Insider's Guide to Warcraft Lore.