Without the use of console commands? You can effectively do anything with console commands.
The only way other than that, if you've already killed him, is if you killed him while already being a "Master Wizard" at the actual time you killed him. (The rank in the guild directly preceding Arch-Mage)
If you weren't, then no, you can't become Arch-Mage.
If you are not opposed to using Console Commands, the easiest thing to do would be
Find his corpse.
Open console commands (~ key on PC, typically)
Click on his corpse.
Type 'resurrect' without the apostrophes.
Press Enter. He should then be alive again. If he attempts to attack you afterwards, open console commands again, referencing him, and enter 'stopcombat'. Then you can bribe him to like you more, and not attack you.
Hope this helps. :)
UPDATE
If you can't find his body, you can also try:
[resurrect trebonius artorius] (without the brackets)
or
["trebonius artorius"->resurrect] (with quotes, no brackets though)
If he still wants to kill you when he sees you,
["trebonius artorius"->setfight 0] (with quotes, no brackets)
If you want to see him immediately after entering this, you can use the placeatme command:
[placeatpc "Trebonius" 20,10,1]
LASTLY
If you want to be cheap about it, and skip straight to completing the guild quest and becoming Archmage, you can
[PCRaiseRank "Mages Guild"] in order to gain a rank
[Journal MG_Guildmaster 100] to complete the quest.
This is a complete list of the joinable factions in Morrowind. A quick Google search of any one of them should be able to help you find any notes regarding how you join them. Don't forget: Joining certain factions closes you off to other ones, and you cannot advance in certain factions beyond a certain point if you've already done a certain amount in another faction. For example, if you've killed the heads of the Thieves Guild for the Fighter's Guild questline, you obviously cannot advance in the Thieves Guild without console commands. There are other NPC factions, and not all these factions will appear in your faction list, but they are all player factions nonetheless.
House Hlaalu
House Redoran
House Telvanni
Blades (Necessary, joined during the main quest)
East Empire Trading Company (Comes with Bloodmoon Expansion)
Fighters Guild
Imperial Cult
Imperial Legion
Mages Guild
Thieves Guild
Ashlander Tribes (Only technically one faction, but becoming friendly with the Ahemmusa, Erabenimsun, Zainabs, and Urshilaku puts you on friendly terms with all the others. You automatically do this during the Trials in the main quest)
Morag Tong (don't listen to the comments that say you cannot join. This is FALSE. You can join by going to their Vivec headquarters in the Arena Canton. In the storage area underneath the arena, talk to Eno Hlaalu. He can initiate you with a quest. This is the only way to join, you cannot join in their guild halls)
Tribunal Temple
Aundae Clan
Berne Clan
Quarra Clan (These are vampire clans. You can only join ONE of them. It will not appear in your faction list, but it will affect the bonuses you receive as a vampire, and what vampire quests are available to you.)
Hope this helped! :D
UPDATE
As for the latter part of your question, if you go to this page on the wiki:
http://www.uesp.net/wiki/Morrowind:Factions
There is a chart down at the bottom that shows the prospective relationships between the guilds, including NPC factions. What that means is that it shows the likelihood to like and/or attack you on sight based on your belonging factions.
Best Answer
According to Fessic's Beginner's Guide to Console Commands, which I found via Google expressly to answer this question, the maximum amount is 65,000. I haven't tested this myself, but it sounds like he knows what he's talking about:
The last note on pressing the up arrow is pretty useful, definitely use that.
A Note on Binary Math
I would not be surprised if the actual number is 65535, which is the 16th power of 2 (minus one). This is the largest possible number that can be stored in a 16-bit unsigned integer.
To support this, note some math:
Truncating that to 16-bits gives
Thus explaining why it has been giving you that particular amount.