Even with the discovery tumor familiar, an alchemist can't normally take the feat Improved Familiar

First, the alchemist would need a for-real familiar. The discovery tumor familiar very carefully never says that the tumor becomes an actual familiar:

The alchemist creates a Diminutive or Tiny tumor on his body, usually on his back or stomach. As a standard action, the alchemist can have the tumor detach itself from his body as a separate creature vaguely resembling a kind of animal suitable for a familiar (bat, cat, and so on) and move about as if it were an independent creature. The tumor can reattach itself to the alchemist as a standard action. The tumor has all the abilities of the animal it resembles (for example, a batlike tumor can fly) and familiar abilities based on the alchemist’s caster level (though some familiar abilities may be useless to an alchemist). The tumor acts as the alchemist’s familiar whether attached or separated (providing a skill bonus, the Alertness feat, and so on). When attached to the alchemist, the tumor has fast healing 5. An alchemist’s extracts and mutagens are considered spells for the purposes of familiar abilities like share spells and deliver touch spells. If a tumor familiar is lost or dies, it can be replaced 1 week later through a specialized procedure that costs 200 gp per alchemist level. The ritual takes 8 hours to complete.

(Emphasis mine.) So the tumor familiar's form is suitable for a familiar, the tumor familiar has the abilities of a familiar, and the tumor familiar acts as a familiar, but the tumor familiar is still technically not a familiar. (Yes, that's terrible.)

Second, although the alchemist can copy the arcane magical writings from a wizard's spellbook, the alchemist doesn't have an arcane caster level. The Alchemist on Alchemy says

Although the alchemist doesn't actually cast spells, he does have a formulae list that determines what extracts he can create. An alchemist can utilize spell-trigger items if the spell appears on his formulae list, but not spell-completion items (unless he uses Use Magic Device to do so). An extract is “cast” by drinking it, as if imbibing a potion—the effects of an extract exactly duplicate the spell upon which its formula is based, save that the spell always affects only the drinking alchemist. The alchemist uses his level as the caster level to determine any effect based on caster level.

(Emphasis mine.) Unfortunately, the alchemist level that the alchemist uses for his extracts is neither arcane nor divine.

If you only want the feat Improved Familiar to make your current familiar celestial, you don't need a feat

Doing so, however, is complicated. This obscure Dungeons and Dragons, Third Edition-era Web-only spell makes this happen:

• The 3rd-level Clr spell planar familiar [trans] (Spellbook Web column "Planar Familiar") instantaneously causes 1 willing touched living animal companion, familiar, special mount, "or other companion" to gain, depending on the caster's patron deity, one of the templates anarchic, axiomatic, celestial, or fiendish. The spell's casting time is 10 min., and casting it costs the caster 500 XP.

For a wizard to cast such a spell, first he must purchase a divine scroll of planar familiar (3rd-level spell at caster level 5) (2,875 gp; 0 lbs.). Then he deciphers the scroll using the 0th-level spell read magic [div] (PH 269) or by succeeding on a Spellcraft skill check (DC 23) or a Use Magic Device skill check (DC 28). If the wizard's Wisdom is less than 13, he must succeed on a Use Magic Device skill check (DC 28) to emulate Wisdom 13. If the wizard's deity isn't appropriate to add the template he wants, the DM may allow success on a Use Magic Device skill check (DC 21) to emulate the level 1 cleric class feature deity, domains, and domain spell, the first part of which says, "Choose a deity for your cleric" (PH 32 and omitted from the SRD). Finally, the wizard casts the spell on his familiar by succeeding on a Use Magic Device skill check (DC 25). Failure means no effect: the scroll still contains the spell, and the wizard can try again. However, a natural 1 on that last Use Magic Device skill check means the wizard must wait until tomorrow to try again.

I told you it was complicated.

A generous DM may house rule that a friendly cleric can cast the spell planar familiar on your behalf, charging you the standard market price of 2,650 gp.

But there are feats (and quests and rituals)

Alternatives include taking, instead, two feats: the feat Extra Familiar (Dragon #280 62) but not gaining this extra familiar until the opportunity presents itself to take the feat Improved Familiar (DMG 200). Further, the Ritual of Vitality or a properly worded wish (both Savage Species 150) may do the trick, but the former is weird and the latter unreliable. Finally, as Ruut mentions, the exalted feat Celestial Familiar (BE 41) allows a creature capable of acquiring a new familiar to gain a celestial familiar, yet dismissing the old familiar remains an issue.

The Player's Handbook II on Class Feature Retraining lists as a class feature that can be retrained by a wizard or sorcerer Choice of familiar (193). Unfortunately, the Process means that you "[c]hange one class feature option [in this case, the choice of familiar] to another legal one[, but t]he new option must represent a choice that you could have made at the same level as you made the original choice" (192), so it's of no help in gaining an improved familiar without suffering from having dismissed or lost the previous familiar.

Voluntarily dismissing a familiar shouldn't cost anything

The 3.5 revision changed for the better the paladin's special mount (which, in Third Edition, was a for-reals horsey that was a source of constant worry because, if it died, a new one was unavailable for (sigh) a year and a day) and the druid's animal companion (which was previously based on the now-obsolete spell animal friendship), but the sorcerer and wizard's familiar suffered a net loss (in Third a wizard could select the toad familiar to gain a +2 bonus to his Constitution score—lots of toad familiars in Third), and, despite the special mount losing its year-and-a-day thing, familiars didn't. Although the sorcerer and wizard don't need any power boosts, I suggest a house rule to even up things anyway because familiars are fun.

House Rules

Pick one.

• A creature can take a full-round action to dismiss its familiar. The creature doesn't lose XP, and by taking 1 day and spending 100 gp he can either gain a new familiar or, if it's still appropriate and present, reinstate the previous one.
• Instead of the class feature summon familiar the sorcerer and wizard gain the special ability urban companion, the wizard's or sorcerer's druid level equaling its wizard or sorcerer level.

I've used both rules (the latter is, obviously, more popular), and my campaigns haven't suffered because of them, but there is a pretty strong gentleman's agreement, and I'm not a DM that routinely threatens familiars.

Without house rules, the only way I know to reduce the wait time between familiars is by taking the feat Wedded to the Light (Dragon #358 87), which, unlike many seemingly similar feats, doesn't have as a prerequisite the ability to summon a new familiar but the ability to summon a familiar, modifying as it does the abilities of the creature's familiar. A good level 13 wizard with that feat automatically spends 200 XP per level but need only wait 1 month to acquire a new familiar after—I kid you not—the wizard dies and his familiar resurrects him by exploding.