Counters
Celebi - This thing pretty much screams counter to Breloom. Its grass/psychic typing gives it resistances to both of Breloom's STABs, which, when combined with its respectable 100/100/100 defensive spread, makes it a nigh insurmountable task for Breloom to break through Celebi. Natural Cure can allow it to heal off Spore, and it can easily outspeed and 1HKO Breloom with Psychic even with minimal Special Attack investment.
Gliscor - Once the Toxic Orb kicks in, Gliscor no longer has to fear Spore from Breloom. He also resists all of Breloom's fighting moves, and has massive defense to weather Breloom's assaults... though repeated hits from Technician Bullet Punch can wear it down eventually, and it doesn't have too much trouble breaking through Gliscor's substitutes. It stops SubPunch Breloom cold, however.
Once Sleep Clause is activated, the number of Pokemon that can counter Breloom goes up dramatically. The following counters do not like being put to sleep, but Sleep Clause helps this matter.
Latios/Latias - The Eon twins resist both of Breloom's STABs, outspeed it with ease, and can 1HKO Breloom with a Psychic/Psyshock or Hidden Power Fire/Ice. As mentioned they don't like eating Spore, but they can revenge kill Breloom with ease if the need arises, as well.
Gengar - With an immunity to Fighting Attacks and a resistance to grass, Gengar doesn't have too much to fear from Breloom. However, its relative frailty means it can't take too many Technician-boosted Bullet Seeds, and said Bullet Seed can rip apart Substitutes. However, SubDisable Gengar can royally screw Breloom over if it Subs on the Spore, and then follows it up by disabling Spore. It goes without saying that eating Spore will ruin Gengar's day, though.
Dragonite - With a 2x resistance to fighting and a 4x resistance to grass, Brelooms not carrying Stone Edge can't touch Dragonite at all. It doesn't like being Spored, of course, but if sleep clause is active, or if Dragonite is packing a Lum Berry when it isn't, it's guaranteed to either set up, or 1HKO Breloom with Fire Punch (possibly both). Adamant Dragonites have to be careful of being outsped by Jolly Brelooms, too.
Salamence - Much like Dragonite, Salamence only fears Stone Edge and Spore, but Intimidate variants can further soften Breloom's attacks up. Once Sleep Clause is in effect, Salamence doesn't mind coming in on most Brelooms.
Dusclops - Same as most of the above; doesn't like being Spored, but once Sleep Clause is active, it's got a fair chance. Its massive Eviolite boosted defenses allow it to survive anything Breloom throws at it, and then cripple it with Will-O-Wisp. Obviously though this only works on Technician Variants.
Skarmory - We all knew this was coming. Skarmory's massive defense can stand up to Breloom's assaults, and in turn Skarmory can 1HKO with Brave Bird. Again, Sleep Clause is your friend as Skarmory doesn't like Spore.
Volcarona - An unconventional counter, as its defense is low, and Stone Edge is guaranteed 1HKO. However, he resists both of Breloom's STABs, and has a chance to burn it with Flame Body, crippling it for the rest of the match. It can also revenge Breloom in a pinch.
Checks
Mamoswine - Jolly Mamoswine can easily revenge kill Breloom with Ice Shard, but it absolutely cannot take a hit from Breloom. For this reason it absolutely has to be Jolly, as a Jolly Breloom will outspeed an Adamant Mamoswine with Mach Punch.
Thundurus-T - It outspeeds Breloom, resists Mach Punch, and 1HKOs with Hidden Power Ice. It doesn't like Spore, or Stone Edge, so it should only be considered as a revenge killer.