i am not the best player, but amumu is my best champion.
here is my guide on how to amumu, which hopefully will enable you to play and build him better.
this is my first and only guide, because this is the champion i feel able to guide anyone.
have fun being stunning,
wantin

amumu is very flexible in his build, but not in his position. imho amumu must jungle. he does not have the strongest early game and he would not be able to farm well mid or top, but he has an incredible jungle clear speed. and is really usefull in teamfights. also he has good ganks. not the best, but good. for amumu support or other break the meta please ask someone else.
amumu offers a lot of cc to the team and can depending on your build deal immense amounts of damage, and be a bit durable, or be quite tanky and do some damage.
he is good at engaging fights, and peeling bruizers from your carries.
last thing to mention here is, that he is an exelent antitank with his w.

i did not get the mechanics of the item thing above also i want to write something about every item so i hope you guys will be fine with this form.

your start is pretty clear. you got to start with a hunters machete and health potions.
i preffer the warding trinket, but if you can make good use of another one pick that.

for your first back try to get at least 760 gold as you need the spirit stone, boots and at least one health pot.

sunfire cape: this is some awesome item for amumu. you will be in the middle of fights anyways since you need to be there, to deal your damage. you have magic penetration with your passive and eventually with runes and other items, which enhances the damage of this item.
it gives you health and armor, which are the first stats you will be looking for in most games. it is an exelent midgame item and i rush it in most games.
as soon as you are level 7 or higher, have a spirit stone and one big armor you can solo dragon, but soloing dragon is risky unless you are sure about the position of all enemies, the jungler especially, so care with doing this. dewarding and then setting up a trap for an enemy or doing drake after successfully ganking bot is often a better idea.
it falls off a bit late, so if you are fullbuild and you feel the need to build something else you might sell this.

spirit of the ancient golem. this is a great item, but you do not need it, what does not mean, that there are a lot of games in which it is worth getting it. it provides you with life cd reduction and will over time get the additional gold back, but it will delay your sunfirecape. It also provides tenacity, which brings me to the next theme:

shoes: in most games i would go for merc treads. at least in the end. other possible choices:

sorcerers boots: if the enemy team has no, or low cc these might be worth it, since you will get 15 magic penetration with the 15-25 from your passive and eventually 8 from runes and eventually 15 from haunting guise this will skyrocket your damage to targets with not too much magic resistance. (normal value would be 30 + 12 for most carry-ish champions unless they get mr) this is also a possibility, if you get spirit of the elder lizard.

mobility boots: handy if you want to be ganking a lot and not jungling toooo much, but since amumu has an exelent clear i do not like them best. if you got them i would trade them in the lategame for merc treads. they are nice early, since they are so cheap now...

ninja tabi: vs very ad heave cc low teams, or cc low teams when you are needed as the main tank.

in general i think the captain enchantment is best on amumu, but homeguards can be needed. get your enchantment late since other items need your money more. also do not forget to upgrade your trinket.

iceborn gauntletts: i love those on amumu. you engage with your stun and you slow, you have your e and w and enemies have a hard time not hugging lonely mumu. it offers armor and a slow also 10% cd and 500 very much needed mana. now we need to start some more topics:

mana: amumu loves bluebuff, but most midlanes do, too. if your midlaner needs blue and you have no mana items yet get some mana potions. most games i will either build an iceborn gauntlett, or a frozen heart. from there on mana is not such a big issue anymore and the cooldown from bluebuff is oftenly not needed anymore, either. spiritstone is also important for mana, so try not to sell it, when you are still jungling a lot and do not jungle too much when you sold it.

auras: there are several items with auras, that mumu can build. talk to your team!
it is not really good to have auraitems more than once in a team (even though aegis stacks for the 2 people that have it, but not for the rest of the team)

slows: i like to have a slow an amumu. but he does not need more than one.

let's move on!

frozen heart: i like the iceborn gauntlett better and building both will result in needlessly much mana and occasionally wasted cd and also often in not enough health. but: against attack speed reliant teams (example: ww top, jax jungle, cho mid, vayne and nami bot) your team should have one, and you can fit it into your build so unless there is someone who will build it instead of you do it.

rylais: offers a slow 500 health and 80 ap. it is nice, but is not really tanky and you do not need it if you have the iceborn gauntlett. but if i get a frozen heart i get this most of the time, too.

lissandrys: i build it in about 90% of my games (in which i reach fullbuild). if you get magic pen from something else early, or if you are ahead you might get spirit visage after some armor early, but most of the times i get it late, since the passive gets stronger lategame. it procs on all of your spell, even the w but not on the sunfire cape. with a slow from rylais or iceborn it gets even stronger, but even without it is worth it.

zonias hourglass: i only build this on rare occasion and most of the time rather late into the game. if you are the main tank this is a no go, but if your team is tanky this is amazing. you can avoid stuff like caits, darius etc ult and you can stop turrets from focussing you (try not to kill your mates by doing that) your w will continue to kill people around you as well as lissandrys and your e has such a low cd, that often you can cast it before and after the hourglass. it gives you some armor and lots of damage since your ap ratios are really good (1.0 on q; something pretty good on w; 0.5 on e, which you spam; 1.0 on ult)

randuins omen: is a good tank item so it is not bad for amumu, but i feel like there are better ones. you might use this one as your slow item if you need to be maintank, since iceborn and rylais are rather offensive items.

locket of the iron solari/aegis of the legion: in most team this item is good to have.
it gives you all defensive stats and 10% cd. if your support and toplaner do not build it, which happens rather often in s4 with all the ap supports you might consider getting it.

banshees veil: awesome! health, which you are craving for, magic resistance, which is nice to have and a passive, that has a huge potential. very good, vs teams, that start of with huge damage or strong cc, not as good vs teams, that destroy the passive with something rather weak.

abysalls szepter: if your team has huge amounts of ap (ap mid, ap support and u) it is good to have one of these in the team. you are one, that defenitly can get it. boosts your and your allies damage, gives you some mr, but offers no health.

spirit visage: this is a very potent item on most tanks, but in most cases not the best to get on amumu, because he has (other than his spirit stone) no healing. nevertheless there can be games in which this would be helpfull since it gives you live, magic resist and 10% cd.

Guardian Angel: on rare occasion i build this. it gives you some magic resist and some armor, but no live. the passive is nice.

warmorgs: i can not recall building this on amumu in quite some time. it got nerfed after i think s2 or so, when everyone built it and since then is mostely seen on champions scaling with health, but i am sure, that it can be nice on amumu and i saw other well playing amumus building it. definetly only build it, if you need to be the main tank.

spirit stone: can be good since wards win games, but it gives you rather few helpfull stats and since trinkets exist it is not essential. i built it last season a lot.

at that topic: if you have a free slot and a little spare money: get wards. wards win games and you walk around the map anyways.

i hope i covered all items.. now we need to combine:
get armor and life early since without life resistances are useless and you need armor since towers, monsters and some champions do normal damage.
do not dive without armor and life and do not dive at low levels or if you are not sure, if you will win the fight then.
get the resistances you need vs the stronger opponents and if the enemy team is not ballanced get more of the resistance, that is needed more (i am just stating all the obvious here...)
do not get auras more than once and do not forget, that more than 40% cd reduction is not possible, so stats would be lost (what does not mean do never build those items, cause sometimes they are still worth). Blue Potions offers 10% cd if you end up with 30%.

and lastly: if you are the tanky one in your team: play as the tanky one and engage and stuff. get the enemys to attack you instead of your mates.
if you are rather offtanky try to only engage, if it is a good oportunity, and your for example mundo will be in the middle of the fight shortly after.
not engaging, but rather keeping enemy bruizers and assassins away from your carries is also an exelent choice.

start blue and do at least one more camp before red to be level 3 before a gank.
or do red or blue and do a lvl 2 gank.
alternatively do blue, wolfs, wraiths, red eventually even golems and then gank, which gives you more xp and gold, which is nice since your early ganks are not the best.
when coming from blue to wolfs go enough to the right to hit the big wolf with the bandage toss. auto attack him before using the e (same with blue) since the magic pen will make your damage bigger. when coming from wolfs to wraiths do not walk to them, but use your bandage toss to the position, where the big one is. (if you have to wait 1 sec till cd wait it since you are still faster, than walking) auto then e and so on. again coming from behind with the bandage toss to red will speed up the process. if you suspect enemy jungler to be in the bush trinket it. people like to counterjungle amumu. if he is there, ask for help, continue your red till help arrives and kill their jungler. try not to let him smite the red.
try not to cancel your autoattacks with your e, but to cast it in between.

amumu can good gank from the side or from behind through bushes, if he is not seen that way and engage with bandage toss.
if you have redbuff do an autoattack after the bandage toss before your e for the slow and the damage.
if you gank from behind in most situations it is better to hold the bandage toss till the run away from you instead of into your direction and instead do damage with w e and autos. or to use your ultimate first. but if the laner needs to stun to catch up use it straight away.
if you use your ultimate and your q try to maximize stun time, so unless you need to cast ult instantly to stun a group do an e first.
in general going your ult first and q second is easier, since not imobilized targets could be able to dodge the q or minions could be in the way, which are often enough killed by your ult which enables you to land your q on the enemy, too.
if you can let the carry have the kill, if he would not get it make it yourself.
in general enemies do not expect you to flash into your ult, which is great for you. but you have no disengage left then so only do that, if you are sure about it.

i hope my guide was able to help you, if you have suggestions how to change this guide, how to play amumu or how to bake bagels, just write me a or leave a comment. same if you have any open questions.
so long,
stay melancholic,
wantin