Jeg kjøpte en sykkel for to uker siden.

kjøpe is a weak (regular) verb in Norwegian Bokmål. For most weak verbs, you form the past (preterite) by adding -te or -et to the stem. Here’s the pattern for kjøpe:
• Infinitive: kjøpe (to buy)
• Present tense: kjøper (buy/buys)
• Past tense: kjøpte (bought)
• Present perfect: har kjøpt (have bought)

Because a specific time in the past is mentioned (“for to uker siden”), you must use the simple past kjøpte rather than the present perfect.

Norwegian nouns have three genders: masculine, feminine, and neuter. In Bokmål:
• Masculine nouns typically take en in the indefinite singular.
• Feminine nouns can take ei (less common in modern Bokmål) or en.
• Neuter nouns take et.

sykkel (“bike”) is treated as a masculine (common) noun in Bokmål, so the correct indefinite article is en: en sykkel.

To talk about more than one week, you need the plural. In Bokmål, most masculine and feminine nouns form the indefinite plural by adding -er to the singular stem. For uke (week):
• Singular indefinite: en uke
• Plural indefinite: uker (weeks)

Thus to uker means “two weeks.”

The full expression for “ago” in Norwegian is for … siden. You need both parts to say “X ago”:
• for = for
• to uker = two weeks
• siden = ago

You cannot leave out for in this construction. (Note: siden alone can mean “since” in other contexts, but not in the “ago” phrase.)

Yes. Norwegian follows the V2 (verb-second) rule in main clauses: if you place an adverbial first, the finite verb must follow immediately, then the subject. So you can also say:
For to uker siden kjøpte jeg en sykkel.
Here:

1. For to uker siden (time adverbial)
2. kjøpte (verb)
3. jeg (subject)
4. en sykkel (object)