Below is a list of all the Background pairings I think jive the best, with asterisks in front of pairings that especially caught my eye (not necessarily all the best ones).  In parentheses are the colors of the cards and the primary themes to build around and/or benefit from.  I find it easiest to use one tab for this list/page, another for the creatures, and another for the enchantment Backgrounds.  Comment if you find another pairing that looks good that I didn't include, or if you think a listed pairing is deserving of an asterisk.

*Abdel Adrian, Gorion's Ward + Far Traveler (W ETB flicker, Soldier tokens)

Abdel Adrian, Gorion's Ward + Veteran Soldier (W Soldier tokens, flicker)

Amber Gristle O'Maul + Agent of the Shadow Thieves (R/B aggro)

Amber Gristle O'Maul + Guild Artisan (R aggro discard/draw, treasures)

Amber Gristle O'Maul + Veteran Soldier (R/W discard/max-draw, Soldier tokens)

*Baeloth Barrityl, Entertainer + Agent of the Iron Throne (R/B Treasures, ramped goad)

*Baeloth Barrityl, Entertainer + Clan Crafter (R/U Treasures, goad, card draw)

Baeloth Barrityl, Entertainer + Folk Hero (R/W Elf/Shaman tribal, treasures, goad)

Baeloth Barrityl, Entertainer + Guild Artisan (R extra treasures, goad)

*Baeloth Barrityl, Entertainer + Noble Heritage (R/W treasures, ramped goad, protection)

*Baeloth Barrityl, Entertainer + Raised by Giants (R/G treasures, goad)

Burakos, Party Leader + Folk Hero (B/W Cleric/Rogue/Warrior/Wizard tribal, treasures, card draw, party)

Burakos, Party Leader + Master Chef (B/G Cleric/Rogue/Warrior/Wizard tribal, treasures, party, +1/+1 counters)

*Durnan of the Yawning Portal + Acolyte of Bahamut (G dragons cost less, creature filtered card draw)

Durnan of the Yawning Portal + Agent of the Shadow Thieves (G/B creature cost reduction)

Durnan of the Yawning Portal + Master Chef (G creatures cost less, +1/+1 counters)

Durnan of the Yawning Portal + Passionate Archaeologist (R/G cast exiled creatures, damage)

Durnan of the Yawning Portal + Tavern Brawler (R/G extra exiled creature/card casts)

Ellyn Harbreeze, Busybody + Inspiring Leader (W creature tokens, filter draw)

Gale, Waterdeep Prodigy + Scion of Halaster (U/B instant/sorcery)

Ganax, Astral Hunter + Acolyte of Bahamut (R/G Dragons, treasures)

Ganax, Astral Hunter + Dragon Cultist (R Dragons, tokens, treasures)

Ganax, Astral Hunter + Feywild Visitor (R/U Dragons, tokens, treasures)

Gut, True Soul Zealot + Agent of the Iron Throne (R/B creature/artifact sac, creature tokens, opp lose life)

Gut, True Soul Zealot + Cloakwood Hermit (R/G creature tokens, creature sac)

Gut, True Soul Zealot + Guild Artisan (R creature & treasure tokens)

Gut, True Soul Zealot + Veteran Soldier (R/W creature tokens)

Jaheira, Friend of the Forest + Cloakwood Hermit (G tokens, ramp)

Jaheira, Friend of the Forest + Guild Artisan (R/G tokens, ramp)

Jaheira, Friend of the Forest + Inspiring Leader (G/W tokens, ramp)

Jaheira, Friend of the Forest + Veteran Soldier (G/W tokens, ramp)

Karlach, Fury of Avernus + Agent of the Shadow Thieves (R/B extra combat, commander protection)

Karlach, Fury of Avernus + Cloakwood Hermit (R/G extra combat, creature tokens)

Karlach, Fury of Avernus + Dragon Cultist (R extra combat, dragon tokens)

Karlach, Fury of Avernus + Hardy Outlander (R/G extra combat, P/T boost)

*Karlach, Fury of Avernus + Haunted One (R/B extra combat, first strike +2/+0, Undying, Tiefling Barbarian (47), +1/+1 counters)

Lae'zel, Vlaakith's Champion + Agent of the Shadow Thieves (W/B +1/+1 counters, tron commander)

Lae'zel, Vlaakith's Champion + Clan Crafter (W/U artifact sac, +1/+1 counters, card draw)

Lae'zel, Vlaakith's Champion + Far Traveler (W counters, flicker, Energy counters?)

Lae'zel, Vlaakith's Champion + Haunted One (W/B Undying Gith/Warrion, +1/+1 counters)

*Lae'zel, Vlaakith's Champion + Master Chef (W/G +1/+1 counters)

Lae'zel, Vlaakith's Champion + Noble Heritage (W +1/+1 counters, protection)

Livaan, Cultist of Tiamat + Dragon Cultist (R instant/sorcery/enchant damage, dragon tokens)

Livaan, Cultist of Tiamat + Passionate Archaeologist (R exile spells = damage, noncreature = +X/+0)

Livaan, Cultist of Tiamat + Scion of Halaster (R/B noncreature spell = +X/+0, library draw sifter)

Livaan, Cultist of Tiamat + Tavern Brawler (R noncreature spell = +X/+0, cast exile card +X/+0)

Lulu, Loyal Hollyphant + Cultist of the Absolute (W/B aggro/tron, creature sac, +1/+1 counters)

Lulu, Loyal Hollyphant + Far Travelever (W +1/+1 flicker, but timing...?)

Lulu, Loyal Hollyphant + Guild Artisan (R/W +1/+1 counters, treasures, aggro)

Lulu, Loyal Hollyphant + Master Chef (W/G +1/+1 counters, flicker...?)

Lulu, Loyal Hollyphant + Popular Entertainer (R/W +1/+1 counters, aggro, goad)

Lulu, Loyal Hollyphant + Street Urchin (R/W +1/+1 counters, sac creature/artifact, ping damage)

Rasaad yn Bashir + Dragon Cultist (R/W P/T swap, dragon tokens)

*Renari, Merchant of Marvels + Acolyte of Bahamut (G/U dragons [66], ramp/flash)

Renari, Merchant of Marvels + Dragon Cultist (R/U dragons, flash, tokens)

Safana, Calimport Cutthroat + Dungeon Delver (B/U dungeon/initiative, treasures)

Sarevok, Deathbringer + Criminal Past (B life loss = power, tron)

Sarevok, Deathbringer + Cultist of the Absolute (B life loss = power, creature sac, tron)

Sarevok, Deathbringer + Folk Hero (W/B life loss, human/knight tribal, card draw)

Sarevok, Deathbringer + Guild Artisan (R/B life loss, treasures)

Sarevok, Deathbringer + Haunted One (B life loss, human/knight tribal, undying, +1/+1 counters)

Sarevok, Deathbringer + Noble Heritage (W/B life loss, tron, protection)

Sarevok, Deathbringer + Raised by Giants (B/G life loss = power 10/10)

Shadowheart, Dark Justiciar + Dragon Cultist (R/B card draw, creature sac, dragon tokens)

Shadowheart, Dark Justiciar + Inspiring Leader (W/B card draw, creature sac, creature tokens)

Shadowheart, Dark Justiciar + Master Chef (B/G card draw, creature sac, +1/+1 counters)

Shadowheart, Dark Justiciar + Noble Heritage (W/B card draw, creature sac, +1/+1 counters, protection)

*Sivriss, Nightmare Speaker + Cloakwood Hermit (B/G card draw/life loss, creature sac, Squirrel tokens)

Sivriss, Nightmare Speaker + Feywild Visitor (U/B card draw/life loss, creature sac, dragon tokens)

Sivriss, Nightmare Speaker + Haunted One (B Undying Snake/Cleric/Warlock)

Skanos Dragonheart + Acolyte of Bahamut (G dragons)

Skanos Dragonheart + Dragon Cultist (R/G dragons, dragon tokens)

Vhal, Candlekeep Researcher + Agent of the Shadow Thieves (U/B non-spell ramp, tron, +1/+1 counters)

Vhal, Candlekeep Researcher + Clan Crafter (U non-spell ramp, tron, artifact sac, +1/+1 counters)

Vhal, Candlekeep Researcher + Cultist of the Absolute (U/B non-spell ramp, tron, creature sac)

Vhal, Candlekeep Researcher + Haunted One (U/B Undying Human/Wizards, non-spell ramp)

Vhal, Candlekeep Researcher + Noble Heritage (W/U non-spell ramp, tron, protection)

Vhal, Candlekeep Researcher + Raised by Giants (U/G non-spell ramp, 10/10, 10 mana)

Viconia, Drow Apostate + Criminal Past (B recursion, creature sac)

Viconia, Drow Apostate + Cultist of the Absolute (B recursion, creature sac, tron)

Viconia, Drow Apostate + Scion of Halaster (B recursion, filter draw)

Volo, Itinerant Scholar + Far Traveler (W/U card draw, ETB, creature flicker)

Volo, Itinerant Scholar + Folk Hero (W/U card draw, human/wizard tribal)

Volo, Itinerant Scholar + Haunted One (U/B Undying Human/Wizard tribal, card draw)

Volo, Itinerant Scholar + Inspiring Leader (W/U card draw, creature tokens)

Volo, Itinerant Scholar + Master Chef (U/G card draw, +1/+1 counters)

Volo, Itinerant Scholar + Tavern Brawler (U/R card draw, exile cast)

Wyll, Blade of Frontiers + Agent of the Shadow Thieves (B/R dice roll, +1/+1 counters, tron)

Wyll, Blade of Frontiers + Clan Crafter (U/R dice roll, +1/+1 counters, tron, card draw, artifact sac)

Wyll, Blade of Frontiers + Cultist of the Absolute (B/R dice roll, tron, creature sac)

Wyll, Blade of Frontiers + Far Traveler (W/R dice roll, tron, flicker)

Wyll, Blade of Frontiers + Folk Hero (W/R dice roll, tron, card draw, human/warlock tribal)

Wyll, Blade of Frontiers + Haunted One (B/R dice roll, human/warlock tribal, undying, ETB)

Wyll, Blade of Frontiers + Scion of Halaster (B/R dice roll, tron, filter draw)

Wyll, Blade of Frontiers + Tavern Brawler (R dice roll, tron, exile cast)

Zellix, Sanity Flayer + Haunted One (U/B Undying Horror, mill)

Zellix, Sanity Flayer + Inspiring Leader (W/U mill, Horror tokens)

Zellix, Sanity Flayer + Master Chef (U/G mill, Horror tokens, +1/+1 counters)

Zellix, Sanity Flayer + Street urchin (U/R mill, Horror tokens, creature/artifact sac = ping damage)