I'm not an expert and this guide should be taken with a grain of salt, comment anythhing incorrect that I wrote. I'm going to extend this guide to add items and more rune choices with indepth descriptions.

as ekko your health is a resourse to be traded for gold and exp from minions, pre level 3 most ranged champs will have an easy time taking more hp than is worth for the gold so scraficing the first bit of cs will help with your major goal of getting a big trade or kill level 3. Against melee champs its harder for them to trade into you without being punished, if its a strong early champ like renekton or yone focus on farming with your q and not giving them an oportunity to all in you. A common stratgie for ekko and most tp champs is to heavily trade with the lane opponent then take a recall and quickly tp back to lane, an example would be against yone you would trade in while keeping your wave 1 minion down, the trades dont have to be won just try to deal damage while staying away from a potential death (special note if youre confident you can try to trade flashes or ignite to make the post recall lane dominance easier)

The most basic description of roaming i can give is leaving lane to gank a side lane, for a roam you want to have shoved your own wave into the enemies mid turret and have some form of vision denial as you head up or down river. Most roams will be to bot lane due to more players and easier enemies to kill. You want to minimise time spent roaming because if you miss out and get nothing the enemy midlaner will normally get higher value from cs and tower plates. The biggest tip is only roam when you're assured more gold or exp than you're giving up. the way to assure a roam is successful is to look at the lane state of where you're roaming, for example if a twitch lulu is being shoved in by a draven leona, leona will most likely have all abilities up so if your lane is down sums or abilities its not a safe roam. Another way to tell is if a lane that is safe to fight has their lane pushing to the friendly tower they will most likely have to overextend to finish crashing the wave and as a result will give you a very easy gank.

You should try your hardest to rotate to fights that aren't obviously impossible. you're a snowballing champ so getting even a little bit of value can get helps start the snowball and helps get your jungler to avoid losing the skirmish. Basically you want to play like galio, rotate to fights in jungle or river as long as you aren't full shoved under turret. For example if you have a tanky jungler like zac you wanna wait to see an opportunity for the enemy to fully commit to trying to kill him, with their abilities and sums being put towards zac you'll have an easy time cleaning up or if your fast enough keeping zac alive.

The general wave managment tip i can give is to try to keep wave on your half of the lane for as much as you can unless you are planning to roam or recall. if while keeping the lane on your half the enemy stacks too many minions and is threatening a dive use your w to zone to have free safe access to clear the wave to the degree necesary to stop the dive.

Starting Items: normally with starting items I'll start dark seal because its the most versitile and scales the best, corrupting pot is good against poke lanes and d ring is probably the worst but is at its best when in a melee matchup.
First Back: i personally aim for a blasting wand on my first back to help with the pure ap and giving an earlier back than hextech.
Boots: build sorc shoes unless your against a burst mage who starts their combo with hard cc (annie) then build mercs so you can have time to use ult before you get killed.
Mythic: protobelt is the best in 99 percent of games i like crown as a 3rd item when im really fed into a bursty team.
Second Item: I personally like lichbane the most but nashor's tooth is the most adaptable, the reason i like lichbane more is the movespeed and extra burst it provides. Shadowflame is optional on second if you're decently fed and against a comp with low mr and/or high shielding.
Third Item: as you reach third item you should have sufficient damage to be able to build defensive with zhonias or banshies. I almost always use zhonias when playing normally or looking to solo carry but if you aren't the solo carry or are against instant cast cc i would suggest banshies to easily 1 for 1 the enemy carry.
Situational items: my go to for after 3rd item is to upgrade to mejais and pick up a rabadons or a void staff but if necessary build into the tankier or more supportive items.
Sixth Item: For your 6th item or boots replacement i like to run deadmans plate to give the movespeed you need to keep your tempo up while giving more stats.

In teamfights identify 2 enemies you can burst and if you're fed wait for one of them to be grouped with a teammate and burst the target while damaging the ally with your aoe. If you arent fed wait till a target is alone or with the other target and you'll be able to 1 shot them. (if your game went poorly enough that there is no one on the enemy team you ca burst focus on ccing at least 2 enemies with w and try to burn their cc with your ult.

When using tp you want to focus on either getting a major lead in lane or minimizing a huge possibe loss in lane. With the current tp (everything is tpable) you can use tp to get to jg skirmishes and tp to wards to gank sidelanes quickly. After lane phase ends you use tp to get to fights if you're busy sidelaning or you can tp to sidelanes if you can get a tower or inhibitor. Generally out of lane phase you want to get as much value from tp as possible from joining teamfight or from getting objectives, a common practice is placing a ward behins where you think the next fight will happen so you can have backline access while sidelaning.